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      <text>17.1.1.12 The Hyperbola</text>
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      <text><![CDATA[<h4>The hyperbola</h4>
<p>The standard form of the hyperbola is denoted (a) in the figure below. This has standard equation </p>
\[{x^2\over a^2}-{y^2\over b^2}=1.\] The eccentricity, \(e\), is defined by \[e^2=1+{b^2\over a^2}\qquad e\gt 1).\]
<figure class="HELM_figure">
    <img src="@@PLUGINFILE@@/17.1%20Hyperbola%20fig1.svg" alt="The standard form of the hyperbola." width="616" height="236" class="img-responsive atto_image_button_text-bottom"></figure>
<p>Note the change in sign compared to the equivalent expressions for the ellipse. The lines \(y=\pm \dfrac{b}{a}x\) are <strong> asymptotes</strong> to the hyperbola (these are the lines to which each branch of the hyperbola approach as \(x\rightarrow \pm\infty\)).</p>
<p>If light is emitted from one focus then on hitting the hyperbolic curve it is reflected in such a way as to appear to be coming from the <em>other</em> focus. See (b) in the above figure. The hyperbola has fewer uses in applications than the other conic
    sections and so we will not dwell here on its properties.<br></p>

<div class="HELM_keypoint">
    <h4>Key Point</h4>
    <p>The standard equation of the hyperbola with foci at \((\pm ae,0)\) is \[\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\qquad \text{with eccentricity}\ e\ \text{given by}\quad\ \ e^2=1+\dfrac{b^2}{a^2}\quad (e\gt;1)\]
</p></div>

<p><br></p>
<h5>General conics</h5>
<p>The conics we have considered above - the ellipse, the parabola and the hyperbola - have all been presented in standard form:- their axes are parallel to either the \(x\)- or \(y\)-axis. However, conics may be rotated to any angle with respect to the axes: they clearly remain conics, but what equations do they have? It can be shown that the equation of <strong>any conic</strong>, can be described by the quadratic expression</p>
    \[Ax^2+Bxy+Cy^2+Dx+Ey+F=0\]
    <p>where \(A,B,C,D,E,F\) are constants.</p>
    <p>If not all of \(A,B,C\) are zero (and \(F\) is a suitable number) the graph of this equation is
    </p>
    <ol type="i">
        <li>an ellipse if \(B^2\lt4AC\)(circle if \(A=C\) <strong>and</strong> \(B=0\)); </li>
        <li>
            a parabola if \(B^2=4AC\);</li>
        <li> a hyperbola if \(B^2\gt 4AC\).</li>
    </ol>
    <hr>
    <h4 class="HELM_example">Example</h4>
    <p>Classify each of the following equations as ellipse, parabola or hyperbola: </p>
    <ol class="HELM_parts_inline">
        <li>\(x^2+2xy+3y^2+x-1=0\)</li>
        <li>\(x^2+2xy+y^2-3y+7=0\)</li>
        <li>\(2x^2+xy+2y^2-2x+3y=6\)</li>
        <li>\(3x^2+2x-5y+3y^2-10=0\)</li>
    </ol>

    <h4 class="HELM_solution">Solution</h4>
    <ol class="HELM_parts">
        <li> Here \(A=1,\ B=2,\ C=3 \qquad \therefore \quad B^2\lt 4AC\). This is an ellipse.</li>
        <li> Here \(A=1,\ B=2,\ C=1 \qquad \therefore \quad B^2=4AC\). This is a parabola.</li>
        <li>Here \(A=2,\ B=1,\ C=2 \qquad \therefore\quad B^2\lt 4AC\) also \(A=C\) but \(B\neq 0\). This is an ellipse.</li>
        <li> Here \(A=3,\ B=0,\ C=3 \qquad \therefore\quad B^2\lt 4AC\). Also \(A=C\) and \(B=0\). This is a circle.</li>
    </ol>]]></text>
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