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<!DOCTYPE html>
<html data-require="math math-format expressions">
<head>
<meta charset="UTF-8" />
<title>Equation of an ellipse</title>
<script src="../khan-exercise.js"></script>
<style type="text/css">
#answer_area .short input[type=text] {
width: 60px;
}
</style>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="A">randRange( 2, 9 )</var>
<var id="B" data-ensure=" A !== B" >randRange( 2, 9 )</var>
<var id="H">randRange( -9, 9 )</var>
<var id="K">randRange( -9, 9 )</var>
<var id="MAJ">max( A, B )</var>
<var id="MIN">min( A, B)</var>
<var id="X2T">H === 0 ? "\\dfrac{x^2}{" + A*A + "}" : "\\dfrac {" + expr(["^", ["+", "x", -H], 2]) + "}{" + A*A +"}" </var>
<var id="Y2T">K === 0 ? "\\dfrac{y^2}{" + B*B + "}" : "\\dfrac {" + expr(["^", ["+", "y", -K], 2]) + "}{" + B*B +"}" </var>
</div>
<div class="problems">
<div>
<div class="question">
<p>The equation of an ellipse <code>E</code> is <code><var>rand(2) == 1 ? expr(["+", Y2T, X2T]) : expr(["+", X2T, Y2T])</var> = 1</code>.</p>
<p>What are its center <code>(h, k)</code> and its major and minor radius?</p>
</div>
<div class="solution" data-type="multiple">
<p class="short"><code>(h, k) = (</code><span class="sol"><var>H</var></span><code>,</code> <span class="sol"><var>K</var></span><code>)</code></p>
<p>Major radius = <span class="sol"><var>MAJ</var></span></p>
<p>Minor radius = <span class="sol"><var>MIN</var></span></p>
</div>
</div>
</div>
<div class="hints">
<p>The equation of an ellipse with center <code>(h, k)</code> is <code> \dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1</code>.</p>
<p>We can rewrite the given equation as <code>\dfrac{(x - <var>negParens(H)</var>)^2}{<var>A*A</var>} + \dfrac{(y - <var>negParens(K)</var>)^2}{<var>B*B</var>} = 1 </code>.</p>
<p>Thus, the center <code>(h, k) = (<var>H</var>, <var>K</var>)</code>.</p>
<p><code><var>MAJ*MAJ</var></code> is bigger than <code><var>MIN*MIN</var></code> so the major radius is <code>\sqrt{<var>MAJ*MAJ</var>} = <var>MAJ</var></code> and the minor radius is <code>\sqrt{<var>MIN*MIN</var>} = <var>MIN</var></code>.</p>
</div>
</div>
</body>
</html>
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