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<!DOCTYPE html>
<html data-require="math math-format expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Equation of a circle in non-factored form</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="H">randRange(-9, 9)</var>
<var id="K">randRange(-9, 9)</var>
<var id="R">randRange(1, 7)</var>
<var id="X2T">H === 0 ? "x^2" : expr(["^", ["+", "x", -H], 2])</var>
<var id="Y2T">K === 0 ? "y^2" : expr(["^", ["+", "y", -K], 2])</var>
<var id="D">-2 * H</var>
<var id="E">-2 * K</var>
<var id="F">H * H + K * K - R * R</var>
</div>
<div class="problems">
<div>
<div class="question">
<p>The equation of a circle <code>C</code> is
<code><var>expr(["+",
"x^2",
"y^2",
D === 0 ? null : ["*", D, "x"],
E === 0 ? null : ["*", E, "y"],
F === 0 ? null : F
])</var> = 0</code>.</p>
<p>What is its center <code>(h, k)</code> and its radius <code>r</code>?</p>
</div>
<div class="solution" data-type="multiple">
<p class="short30"><code>(h, k) = (</code><span class="sol" data-forms="integer"><var>H</var></span><code>,</code> <span class="sol" data-forms="integer"><var>K</var></span><code>)</code></p>
<p><code>r = {}</code><span class="sol" data-forms="integer"><var>R</var></span></p>
</div>
</div>
</div>
<div class="hints">
<p>To find the equation in standard form, complete the square.</p>
<p><code>(<var>expr(["+", "x^2", ["*", D, "x"]])</var>) + (<var>expr(["+", "y^2", ["*", E, "y"]])</var>) = <var>-F</var></code></p>
<p><code>(<var>expr(["+", "x^2", ["*", D, "x"], H * H])</var>) + (<var>expr(["+", "y^2", ["*", E, "y"], K * K])</var>) = <var>-F</var> + <var>H * H</var> + <var>K * K</var></code></p>
<p><code><var>X2T</var> + <var>Y2T</var> = <var>R * R</var> = <var>R</var>^2</code></p>
<p>Thus, <code>(h, k) = (<var>H</var>, <var>K</var>)</code> and <code>r = <var>R</var></code>.</p>
</div>
</div>
</body>
</html>
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