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<!DOCTYPE html>
<html>
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<meta charset="UTF-8">
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<title>Ellipses</title>
</head>
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<div id="book-content">
<div class="section" id="fwk-redden-ch08_s03" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">8.3</span> Ellipses</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch08_s03_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch08_s03_o01" numeration="arabic">
<li>Graph an ellipse in standard form.</li>
<li>Determine the equation of an ellipse given its graph.</li>
<li>Rewrite the equation of an ellipse in standard form.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch08_s03_s01" version="5.0" lang="en">
<h2 class="title editable block">The Ellipse in Standard Form</h2>
<p class="para block" id="fwk-redden-ch08_s03_s01_p01">An <span class="margin_term"><a class="glossterm">ellipse</a><span class="glossdef">The set of points in a plane whose distances from two fixed points have a sum that is equal to a positive constant.</span></span> is the set of points in a plane whose distances from two fixed points, called foci, have a sum that is equal to a positive constant. In other words, if points <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0603" display="inline"><mrow><msub><mi>F</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0604" display="inline"><mrow><msub><mi>F</mi><mn>2</mn></msub></mrow></math></span> are the foci (plural of focus) and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0605" display="inline"><mi>d</mi></math></span> is some given positive constant then <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0606" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> is a point on the ellipse if <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0607" display="inline"><mrow><mi>d</mi><mo>=</mo><msub><mi>d</mi><mn>1</mn></msub><mo>+</mo><msub><mi>d</mi><mn>2</mn></msub></mrow></math></span> as pictured below:</p>
<div class="informalfigure large block">
<img src="section_11/6d8fca9d3d015398847484153ddfbe28.png">
</div>
<p class="para editable block" id="fwk-redden-ch08_s03_s01_p03">In addition, an ellipse can be formed by the intersection of a cone with an oblique plane that is not parallel to the side of the cone and does not intersect the base of the cone. Points on this oval shape where the distance between them is at a maximum are called <span class="margin_term"><a class="glossterm">vertices</a><span class="glossdef">Points on the ellipse that mark the endpoints of the major axis.</span></span> and define the <span class="margin_term"><a class="glossterm">major axis</a><span class="glossdef">The line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is a maximum.</span></span>. The center of an ellipse is the midpoint between the vertices. The <span class="margin_term"><a class="glossterm">minor axis</a><span class="glossdef">The line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is a minimum.</span></span> is the line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is at a minimum. The endpoints of the minor axis are called <span class="margin_term"><a class="glossterm">co-vertices</a><span class="glossdef">Points on the ellipse that mark the endpoints of the minor axis.</span></span>.</p>
<div class="informalfigure large block">
<img src="section_11/5f45258d85baa5d81551db000ae472ce.png">
</div>
<p class="para block" id="fwk-redden-ch08_s03_s01_p05">If the major axis of an ellipse is parallel to the <em class="emphasis">x</em>-axis in a rectangular coordinate plane, we say that the ellipse is horizontal. If the major axis is parallel to the <em class="emphasis">y</em>-axis, we say that the ellipse is vertical. In this section, we are only concerned with sketching these two types of ellipses. However, the ellipse has many real-world applications and further research on this rich subject is encouraged. In a rectangular coordinate plane, where the center of a horizontal ellipse is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0608" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span>, we have</p>
<div class="informalfigure large block">
<img src="section_11/58683e5be8efbb778427b0f769d578ff.png">
</div>
<p class="para block" id="fwk-redden-ch08_s03_s01_p08">As pictured <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0609" display="inline"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></math></span> where <em class="emphasis">a</em>, one-half of the length of the major axis, is called the <span class="margin_term"><a class="glossterm">major radius</a><span class="glossdef">One-half of the length of the major axis.</span></span>. And <em class="emphasis">b</em>, one-half of the length of the minor axis, is called the <span class="margin_term"><a class="glossterm">minor radius</a><span class="glossdef">One-half of the length of the minor axis.</span></span>. The equation of an <span class="margin_term"><a class="glossterm">ellipse in standard form</a><span class="glossdef">The equation of an ellipse written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0610" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span> The center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0611" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> and the larger of <em class="emphasis">a</em> and <em class="emphasis">b</em> is the major radius and the smaller is the minor radius.</span></span> follows:</p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p09"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0612" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p10">The vertices are <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0613" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>±</mo><mi>a</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0614" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi><mo>±</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span> and the orientation depends on <em class="emphasis">a</em> and <em class="emphasis">b</em>. If <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0615" display="inline"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></math></span>, then the ellipse is horizontal as shown above and if <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0616" display="inline"><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow></math></span>, then the ellipse is vertical and <em class="emphasis">b</em> becomes the major radius. What do you think happens when <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0617" display="inline"><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></math></span>?</p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p11"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Equation</p></th>
<th align="center"><p class="para">Center</p></th>
<th align="center"><p class="para"><em class="emphasis">a</em></p></th>
<th align="center"><p class="para"><em class="emphasis">b</em></p></th>
<th align="center"><p class="para">Orientation</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0618" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0619" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0620" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>2</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0621" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>3</mn></mrow></math></span></p></td>
<td align="center"><p class="para">Vertical</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0622" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0623" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0624" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0625" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>4</mn></mrow></math></span></p></td>
<td align="center"><p class="para">Vertical</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0626" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>1</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0627" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0628" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0629" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></math></span></p></td>
<td align="center"><p class="para">Vertical</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0630" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>10</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0631" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0632" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>5</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0633" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow></math></span></p></td>
<td align="center"><p class="para">Horizontal</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch08_s03_s01_p12">The graph of an ellipse is completely determined by its center, orientation, major radius, and minor radius, all of which can be determined from its equation written in standard from.</p>
<div class="callout block" id="fwk-redden-ch08_s03_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch08_s03_s01_p13">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0634" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p14">Written in this form we can see that the center of the ellipse is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0635" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0636" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>4</mn></msqrt><mo>=</mo><mn>2</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0637" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mrow><mn>25</mn></mrow></msqrt><mo>=</mo><mn>5</mn></mrow><mo>.</mo></math></span> From the center mark points 2 units to the left and right and 5 units up and down.</p>
<div class="informalfigure large">
<img src="section_11/b1155958685dc952ad66cf47244f7496.png">
</div>
<p class="para" id="fwk-redden-ch08_s03_s01_p16">Then draw an ellipse through these four points.</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p17">Answer:</p>
<div class="informalfigure large">
<img src="section_11/df1388563e6e9256a489bdf92209a12f.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch08_s03_s01_p18">As with any graph, we are interested in finding the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts.</p>
<div class="callout block" id="fwk-redden-ch08_s03_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch08_s03_s01_p19">Find the intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0638" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p20">To find the <em class="emphasis">x</em>-intercepts set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0639" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span>:</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0640" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>4</mn><mrow><mn>25</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mo>−</mo><mfrac><mn>4</mn><mrow><mn>25</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>21</mn></mrow><mrow><mn>25</mn></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s01_p22">At this point we extract the root by applying the square root property.</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0641" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mn>2</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>±</mo><msqrt><mrow><mfrac><mrow><mn>21</mn></mrow><mrow><mn>25</mn></mrow></mfrac></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>±</mo><mfrac><mrow><mn>2</mn><msqrt><mrow><mn>21</mn></mrow></msqrt></mrow><mn>5</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><mo>±</mo><mfrac><mrow><mn>2</mn><msqrt><mrow><mn>21</mn></mrow></msqrt></mrow><mn>5</mn></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>15</mn><mo>±</mo><mn>2</mn><msqrt><mrow><mn>21</mn></mrow></msqrt></mrow><mn>5</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s01_p24">Setting <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0642" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span> and solving for <em class="emphasis">y</em> leads to complex solutions, therefore, there are no <em class="emphasis">y</em>-intercepts. This is left as an exercise.</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p25">Answer: <em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0643" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>15</mn><mo>±</mo><mn>2</mn><msqrt><mrow><mn>21</mn></mrow></msqrt></mrow><mn>5</mn></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercepts: none.</p>
</div>
<p class="para editable block" id="fwk-redden-ch08_s03_s01_p26">Unlike a circle, standard form for an ellipse requires a 1 on one side of its equation.</p>
<div class="callout block" id="fwk-redden-ch08_s03_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch08_s03_s01_p27">Graph and label the intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0644" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p28">To obtain standard form, with 1 on the right side, divide both sides by 9.</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0645" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mstyle color="#007fbf"><mn>9</mn></mstyle></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>9</mn><mstyle color="#007fbf"><mn>9</mn></mstyle></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>9</mn><mn>9</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>1</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s01_p30">Therefore, the center of the ellipse is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0646" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0647" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>9</mn></msqrt><mo>=</mo><mn>3</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0648" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>1</mn></msqrt><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span> The graph follows:</p>
<div class="informalfigure large">
<img src="section_11/7fd602ea4e0b29fc03ba0503785c5257.png">
</div>
<p class="para" id="fwk-redden-ch08_s03_s01_p32">To find the intercepts we can use the standard form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0649" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span>:</p>
<p class="para" id="fwk-redden-ch08_s03_s01_p33"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">x</em>-intercepts set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0650" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><em class="emphasis">y</em>-intercepts set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0651" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0652" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0653" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>4</mn><mn>9</mn></mfrac><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>5</mn><mn>9</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi><mo>−</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>±</mo><msqrt><mrow><mfrac><mn>5</mn><mn>9</mn></mfrac></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mo>±</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt></mrow><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><msqrt><mn>5</mn></msqrt></mrow><mn>3</mn></mfrac></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch08_s03_s01_p34">Therefore the <em class="emphasis">x</em>-intercept is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0654" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> and the <em class="emphasis">y</em>-intercepts are <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0655" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mn>3</mn><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0656" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mn>3</mn><mo>−</mo><msqrt><mn>5</mn></msqrt></mrow><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s01_p35">Answer:</p>
<div class="informalfigure large">
<img src="section_11/ace1155e2a64caca138e4534aafcc7a4.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch08_s03_s01_p36">Consider the ellipse centered at the origin,</p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p37"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0657" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch08_s03_s01_p38">Given this equation we can write,</p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p39"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0658" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mn>1</mn><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p40">In this form, it is clear that the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0659" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0660" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0661" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span> Furthermore, if we solve for <em class="emphasis">y</em> we obtain two functions:</p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p41"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0662" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>±</mo><msqrt><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>±</mo><mn>2</mn><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch08_s03_s01_p42">The function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0663" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>2</mn><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span> is the top half of the ellipse and the function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0664" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span> is the bottom half.</p>
<div class="informalfigure large block">
<img src="section_11/97f453a8d499cbd07f33966ff24551e2.png">
</div>
<div class="callout block" id="fwk-redden-ch08_s03_s01_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch08_s03_s01_p44"><strong class="emphasis bold">Try this!</strong> Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0665" display="inline"><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>36</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s01_p45">Answer:</p>
<div class="informalfigure large">
<img src="section_11/f3cdf60f220bf36bd33288cebf6a47d3.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/2om0ssfRCf4" condition="http://img.youtube.com/vi/2om0ssfRCf4/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/2om0ssfRCf4" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch08_s03_s02" version="5.0" lang="en">
<h2 class="title editable block">The Ellipse in General Form</h2>
<p class="para block" id="fwk-redden-ch08_s03_s02_p01">We have seen that the graph of an ellipse is completely determined by its center, orientation, major radius, and minor radius; which can be read from its equation in standard form. However, the equation is not always given in standard form. The equation of an <span class="margin_term"><a class="glossterm">ellipse in general form</a><span class="glossdef">The equation of an ellipse written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0666" display="inline"><mrow><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi><mo>=</mo><mn>0</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0667" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span> follows,</p>
<p class="para block" id="fwk-redden-ch08_s03_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0668" display="block"><mrow><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi><mo>=</mo><mn>0</mn></mrow></math></span>
where <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0669" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow><mo>.</mo></math></span> The steps for graphing an ellipse given its equation in general form are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch08_s03_s02_n01">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch08_s03_s02_p04">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0670" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>−</mo><mn>90</mn><mi>y</mi><mo>+</mo><mn>239</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s03_s02_p05">Begin by rewriting the equation in standard form.</p>
<ul class="itemizedlist" id="fwk-redden-ch08_s03_s02_l01" mark="none">
<li>
<p class="para"><strong class="emphasis bold">Step 1:</strong> Group the terms with the same variables and move the constant to the right side. Factor so that the leading coefficient of each grouping is 1.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0671" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>−</mo><mn>90</mn><mi>y</mi><mo>+</mo><mn>239</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>90</mn><mi>y</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>239</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>+</mo><mn>9</mn><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>y</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>239</mn></mtd></mtr></mtable></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 2:</strong> Complete the square for each grouping. In this case, for the terms involving <em class="emphasis">x</em> use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0672" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>8</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mn>4</mn><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow></math></span> and for the terms involving <em class="emphasis">y</em> use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0673" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>10</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>25</mn></mrow><mo>.</mo></math></span> The factor in front of the grouping affects the value used to balance the equation on the right side:</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0674" display="block"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>16</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>9</mn><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>y</mi><mstyle color="#007f3f"><mo>+</mo></mstyle><mstyle color="#007f3f"><mn>25</mn></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>239</mn><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>32</mn></mstyle><mstyle color="#007f3f"><mo>+</mo></mstyle><mstyle color="#007f3f"><mn>225</mn></mstyle></mrow></math></span></p>
<p class="para">Because of the distributive property, adding 16 inside of the first grouping is equivalent to adding <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0675" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>16</mn><mo>=</mo><mn>32</mn></mrow><mo>.</mo></math></span> Similarly, adding 25 inside of the second grouping is equivalent to adding <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0676" display="inline"><mrow><mn>9</mn><mo>⋅</mo><mn>25</mn><mo>=</mo><mn>225</mn></mrow><mo>.</mo></math></span> Now factor and then divide to obtain 1 on the right side.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0677" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>18</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mstyle color="#007fbf"><mn>18</mn></mstyle></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>18</mn></mrow><mrow><mstyle color="#007fbf"><mn>18</mn></mstyle></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>18</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>18</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>18</mn></mrow><mrow><mn>18</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>2</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
</li>
<li>
<strong class="emphasis bold">Step 3:</strong> Determine the center, <em class="emphasis">a</em>, and <em class="emphasis">b</em>. In this case, the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0678" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0679" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>9</mn></msqrt><mo>=</mo><mn>3</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0680" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow><mo>.</mo></math></span>
</li>
<li>
<strong class="emphasis bold">Step 4:</strong> Use <em class="emphasis">a</em> to mark the vertices left and right of the center, use <em class="emphasis">b</em> to mark the vertices up and down from the center, and then sketch the graph. In this case, the vertices along the minor axes <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0681" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>±</mo><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span> are not apparent and should be labeled.</li>
</ul>
<div class="informalfigure large">
<img src="section_11/5475698735db634a98d992b5439187bd.png">
</div>
<p class="para" id="fwk-redden-ch08_s03_s02_p07">Answer:</p>
<div class="informalfigure large">
<img src="section_11/aa43277c4cc3f487e8f8f944a8983645.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch08_s03_s02_n02">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch08_s03_s02_p08">Determine the center of the ellipse as well as the lengths of the major and minor axes: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0682" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>40</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s03_s02_p09">In this example, we only need to complete the square for the terms involving <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch08_s03_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0683" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>40</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>30</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>40</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>40</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s02_p11">Use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0684" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>6</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math></span> for the first grouping to be balanced by <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0685" display="inline"><mrow><mn>5</mn><mo>⋅</mo><mn>9</mn><mo>=</mo><mn>45</mn></mrow></math></span> on the right side.</p>
<p class="para" id="fwk-redden-ch08_s03_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0686" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>9</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>40</mn><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>45</mn></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mstyle color="#007fbf"><mn>5</mn></mstyle></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>5</mn><mstyle color="#007fbf"><mn>5</mn></mstyle></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>1</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>5</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s02_p13">Here, the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0687" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0688" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>1</mn></msqrt><mo>=</mo><mn>1</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0689" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>5</mn></msqrt></mrow><mo>.</mo></math></span> Because <em class="emphasis">b</em> is larger than <em class="emphasis">a</em>, the length of the major axis is 2<em class="emphasis">b</em> and the length of the minor axis is 2<em class="emphasis">a</em>.</p>
<p class="para" id="fwk-redden-ch08_s03_s02_p14">Answer: Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0690" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; major axis: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0691" display="inline"><mrow><mn>2</mn><msqrt><mn>5</mn></msqrt></mrow></math></span> units; minor axis: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0692" display="inline"><mn>2</mn></math></span> units.</p>
</div>
<div class="callout block" id="fwk-redden-ch08_s03_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch08_s03_s02_p15"><strong class="emphasis bold">Try this!</strong> Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0693" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>−</mo><mn>16</mn><mi>y</mi><mo>+</mo><mn>25</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch08_s03_s02_p16">Answer:</p>
<div class="informalfigure large">
<img src="section_11/e023b206b176f238df44a742d870dcab.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/Mhp7S5H2820" condition="http://img.youtube.com/vi/Mhp7S5H2820/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/Mhp7S5H2820" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch08_s03_s02_n03">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch08_s03_s02_l02" mark="bullet">
<li>The graph of an ellipse is completely determined by its center, orientation, major radius, and minor radius.</li>
<li>The center, orientation, major radius, and minor radius are apparent if the equation of an ellipse is given in standard form: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0694" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span>
</li>
<li>To graph an ellipse, mark points <em class="emphasis">a</em> units left and right from the center and points <em class="emphasis">b</em> units up and down from the center. Draw an ellipse through these points.</li>
<li>The orientation of an ellipse is determined by <em class="emphasis">a</em> and <em class="emphasis">b</em>. If <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0695" display="inline"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></math></span> then the ellipse is wider than it is tall and is considered to be a horizontal ellipse. If <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0696" display="inline"><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow></math></span> then the ellipse is taller than it is wide and is considered to be a vertical ellipse.</li>
<li>If the equation of an ellipse is given in general form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0697" display="inline"><mrow><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi><mo>=</mo><mn>0</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0698" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span>, group the terms with the same variables, and complete the square for both groupings.</li>
<li>We recognize the equation of an ellipse if it is quadratic in both <em class="emphasis">x</em> and <em class="emphasis">y</em> and the coefficients of each square term have the same sign.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch08_s03_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd01">
<h3 class="title">Part A: The Ellipse in Standard Form</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch08_s03_qs01_p01"><strong class="emphasis bold">Given the equation of an ellipse in standard form, determine its center, orientation, major radius, and minor radius.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0699" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>49</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0703" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>64</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0707" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0712" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0717" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>36</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0721" display="inline"><mrow><mn>16</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>48</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd01_qd02" start="7">
<p class="para" id="fwk-redden-ch08_s03_qs01_p14"><strong class="emphasis bold">Determine the standard form for the equation of an ellipse given the following information.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p15">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0726" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span> with <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0727" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>5</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0728" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p17">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0730" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></math></span> with <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0731" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>7</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0732" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p19">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0734" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span> with <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0735" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>6</mn></msqrt></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0736" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p21">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0738" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>7</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> with <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0739" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0740" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>7</mn></msqrt></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p23">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0742" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> with <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0743" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0744" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>5</mn></msqrt></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p25">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0746" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> with <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0747" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0748" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd01_qd03" start="13">
<p class="para" id="fwk-redden-ch08_s03_qs01_p27"><strong class="emphasis bold">Graph.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0750" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0751" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0752" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>1</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0753" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0754" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>64</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0755" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>49</mn></mrow></mfrac><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0756" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>36</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0757" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0758" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>100</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0759" display="inline"><mrow><mn>81</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>81</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0760" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0761" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>12</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0762" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0763" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>18</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>3</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0764" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0765" display="inline"><mrow><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0766" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0767" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>50</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd01_qd04" start="31">
<p class="para" id="fwk-redden-ch08_s03_qs01_p64"><strong class="emphasis bold">Find the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0768" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0770" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0772" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0775" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0778" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0780" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>72</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0783" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0786" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd01_qd05" start="39">
<p class="para" id="fwk-redden-ch08_s03_qs01_p81"><strong class="emphasis bold">Find the equation of the ellipse.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p82">Ellipse with vertices <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0789" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>±</mo><mn>5</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0790" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>±</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p84">Ellipse whose major axis has vertices <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0792" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0793" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span> and minor axis has vertices <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0794" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0795" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p86">Ellipse whose major axis has vertices <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0797" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>8</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0798" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> and minor axis has a length of 4 units.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p88">Ellipse whose major axis has vertices <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0800" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0801" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span> and minor axis has a length of 2 units.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd02">
<h3 class="title">Part B: The Ellipse in General Form</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd02_qd01" start="43">
<p class="para" id="fwk-redden-ch08_s03_qs01_p90"><strong class="emphasis bold">Rewrite in standard form and graph.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0803" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>36</mn><mi>y</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0805" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>100</mn><mi>y</mi><mo>−</mo><mn>116</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0807" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>49</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>x</mi><mo>+</mo><mn>98</mn><mi>y</mi><mo>−</mo><mn>111</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0809" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>72</mn><mi>x</mi><mo>+</mo><mn>24</mn><mi>y</mi><mo>+</mo><mn>144</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0811" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>64</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>128</mn><mi>y</mi><mo>+</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0813" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>96</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>132</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0815" display="inline"><mrow><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>40</mn><mi>y</mi><mo>−</mo><mn>44</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0817" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0819" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>36</mn><mi>y</mi><mo>−</mo><mn>41</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0821" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>160</mn><mi>x</mi><mo>−</mo><mn>10</mn><mi>y</mi><mo>+</mo><mn>361</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0823" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>32</mn><mi>x</mi><mo>−</mo><mn>20</mn><mi>y</mi><mo>+</mo><mn>64</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0825" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>30</mn><mi>y</mi><mo>+</mo><mn>65</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0827" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>−</mo><mn>27</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0829" display="inline"><mrow><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>28</mn><mi>x</mi><mo>−</mo><mn>16</mn><mi>y</mi><mo>+</mo><mn>46</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0831" display="inline"><mrow><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn><mi>x</mi><mo>−</mo><mn>32</mn><mi>y</mi><mo>−</mo><mn>119</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0833" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>100</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>64</mn><mi>x</mi><mo>−</mo><mn>300</mn><mi>y</mi><mo>−</mo><mn>111</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p123"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0835" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>20</mn><mi>y</mi><mo>+</mo><mn>21</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p125"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0837" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd02_qd02" start="61">
<p class="para" id="fwk-redden-ch08_s03_qs01_p127"><strong class="emphasis bold">Given general form determine the intercepts.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0839" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>20</mn><mi>x</mi><mo>+</mo><mn>24</mn><mi>y</mi><mo>+</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0841" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0845" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0848" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>7</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p136"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0852" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>20</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>18</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0854" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd02_qd03" start="67">
<p class="para" id="fwk-redden-ch08_s03_qs01_p140"><strong class="emphasis bold">Determine the area of the ellipse. (The area of an ellipse is given by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0857" display="inline"><mrow><mi>A</mi><mo>=</mo><mi>π</mi><mi>a</mi><mi>b</mi></mrow></math></span>, where <em class="emphasis">a</em> and <em class="emphasis">b</em> are the lengths of the major radius and the minor radius.)</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0858" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0860" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>18</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p145"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0862" display="inline"><mrow><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>+</mo><mn>36</mn><mi>y</mi><mo>+</mo><mn>94</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0864" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd02_qd04" start="71">
<p class="para" id="fwk-redden-ch08_s03_qs01_p149"><strong class="emphasis bold">Given the graph of an ellipse, determine its equation in general form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa71">
<div class="question">
<div class="informalfigure large">
<img src="section_11/77d9ec6f401101fb60445904c2b9e5cf.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa72">
<div class="question">
<div class="informalfigure large">
<img src="section_11/ceaac1c27ebfdacf70c398411757869c.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa73">
<div class="question">
<div class="informalfigure large">
<img src="section_11/cf3ead1d53975e6c36e7b4995c1f92f3.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa74">
<div class="question">
<div class="informalfigure large">
<img src="section_11/c0a66b6a01cd2a053bab9800d805b634.png">
</div>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s03_qs01_qd03_qd01" start="75">
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p158">Explain why a circle can be thought of as a very special ellipse.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p159">Make up your own equation of an ellipse, write it in general form and graph it.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p160">Do all ellipses have intercepts? What are the possible numbers of intercepts for an ellipse? Explain.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch08_s03_qs01_p161">Research and discuss real-world examples of ellipses.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch08_s03_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p03_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0700" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>; orientation: vertical; major radius: 7 units; minor radius: 2 units; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0701" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>2</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0702" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p07_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0708" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></math></span>; orientation: horizontal; major radius: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0709" display="inline"><mrow><msqrt><mn>3</mn></msqrt></mrow></math></span> units; minor radius: 1 unit; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0710" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0711" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p11_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0718" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span>; orientation: horizontal; major radius: 3 units; minor radius: 2 units; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0719" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0720" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p16_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0729" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p20_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0737" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>6</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>12</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p24_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0745" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa13_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/b53958ca303afa5b0c216db50472aebb.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa15_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/9e9f5466ae3f031b917897e4315b4816.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa17_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/9550eaa6151dedb4804fa71c8162d62b.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa19_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/a8dfca7c363ba28d06abed6999e569ec.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa21_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/4c700e9ed1a845edbac9eb60275b3202.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa23_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/ed88c51ce575a7dc93ed701c40cfd9ea.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa25_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/332680ad079811d8813e1bb55d43f296.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa27_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/bd9b328604d51df0b1c3e2f694d572b5.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa28_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa29_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/bed8a3eb995e00138c387e98b8da9cc8.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa30_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p66_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0769" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>9</mn><mo>±</mo><mn>2</mn><msqrt><mn>5</mn></msqrt></mrow><mn>3</mn></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercepts: none</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa32_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa33_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p70_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0773" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0774" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa34_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p74_ans"><em class="emphasis">x</em>-intercepts: none; <em class="emphasis">y</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0779" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4</mn><mo>±</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa36_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa37_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p78_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0784" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>±</mo><msqrt><mn>2</mn></msqrt><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0785" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>±</mo><msqrt><mn>5</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa38_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa39_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p83_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0791" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa40_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa41_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p87_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0799" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa42_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="43">
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa43_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p92_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0804" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/5bd9952171df55c084e928d02b08eb44.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa44_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa45_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p96_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0808" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>49</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/a9cb7f3ac696205a2b2bcb127773e732.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa46_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa47_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p100_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0812" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>64</mn></mrow></mfrac><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/bddde005efa3182e493c27ff31bf1287.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa48_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa49_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p104_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0816" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/167ea4159775aea700b99fc79ec30dce.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa50_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa51_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p108_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0820" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>81</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/d6bcee76aa5ce3f1a490200e1b772b3b.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa52_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa53_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p112_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0824" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/77a4c994e2d4b186bdc90501bb222350.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa54_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa55_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p116_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0828" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/e68d7473c5e235db17f8697762822b6a.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa56_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s03_qs01_qa57_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s03_qs01_p120_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0832" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/f782675735c2a1f76923e6003805f093.png">
</div>
</div>