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<html data-require="math math-format expressions">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Equation of a hyperbola</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
<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="A2">A * A</var>
<var id="B2">B * B</var>
<var id="H">randRange( -9, 9 )</var>
<var id="K">randRange( -9, 9 )</var>
<var id="WHICH_NEG">rand(2)</var>
<var id="X_MINUS">"+"</var>
<var id="Y_MINUS">WHICH_NEG === 1 ? "" : "-"</var>
<var id="X">H === 0 ? "x^2" : expr(["^", ["+", "x", -H], 2])</var>
<var id="Y">K === 0 ? "y^2" : expr(["^", ["+", "y", -K], 2])</var>
<var id="X2T">H === 0 ? "\\dfrac{x^2}{" + A2 + "}" : "\\dfrac {" + expr(["^", ["+", "x", -H], 2]) + "}{" + A2 +"}" </var>
<var id="Y2T">K === 0 ? "\\dfrac{y^2}{" + B2 + "}" : "\\dfrac {" + expr(["^", ["+", "y", -K], 2]) + "}{" + B2 +"}" </var>
<div class="problems">
<div class="question">
<p>The equation of hyperbola <code>H</code> is <code><var>WHICH_NEG === 1 ? expr(["-", Y2T, X2T]) : expr(["-", X2T, Y2T])</var> = 1</code>.</p>
<p>What are the asymptotes?</p>
<div class="solution" data-type="multiple">
<code>y = \pm </code><span class="sol short28" data-fallback="1"><var>B/A</var></span>
<code>(x + </code><span class="sol short28" data-fallback="0"><var>-H</var></span><code>) + </code><span class="sol short28" data-fallback="0"><var>K</var></span>
<div class="hints">
<p>We want to rewrite the equation in terms of <code>y</code>, so start off by moving the <code>y</code> terms to one side:</p>
<p><code><var>Y2T</var> = <var>Y_MINUS</var> 1 <var>X_MINUS</var> <var>X2T</var></code></p>
<p>Multiply both sides of the equation by <code><var>B2</var></code>.</p>
<p><code><var> Y</var> = {<var>Y_MINUS</var> <var>B2</var> <var>X_MINUS</var> \dfrac{<var>X</var> \cdot <var>B2</var>}{<var>A2</var>}}</code></p>
<p>Take the square root of both sides.</p>
<p><code>\sqrt{<var>Y</var>} = \sqrt{<var>Y_MINUS</var> <var>B2</var> <var>X_MINUS</var> \dfrac{<var>X</var> \cdot <var>B2</var>}{<var>A2</var>}}</code></p>
<p><code><var>plus("y", -K)</var> = \pm \sqrt{<var>Y_MINUS</var> <var>B2</var> <var>X_MINUS</var> \dfrac{<var>X</var> \cdot <var>B2</var>}{<var>A2</var>}}</code></p>
<p>As <code>x</code> approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it.</p>
<p><code><var>plus("y", -K)</var> \approx \pm \sqrt{\dfrac{<var>X</var> \cdot <var>B2</var>}{<var>A2</var>}}</code></p>
<p><code><var>plus("y", -K)</var> \approx \pm \left(\dfrac{<var>B</var> \cdot (<var>plus("x", -H)</var>)}{<var>A</var>}\right)</code></p>
<span data-if="K > 0">Add <code><var>K</var></code> to both sides and rewrite</span>
<span data-else-if="K &lt; 0">Subtract <code><var>-K</var></code> from both sides and rewrite</span>
<span data-else="">Rewrite</span>
as an equality in terms of <code>y</code> to get the equation of the asymptotes:
<p><code>y = \pm <var>fractionReduce(B, A)</var>(<var>plus( "x", -H )</var>)<var>K &gt;= 0 ? "+" : ""</var> <var>K</var></code></p>