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      <text>17.3.3.1 Parametric Form of the Parabola</text>
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      <text><![CDATA[<h4> Standard Forms of Conic Sections in Parametric Form </h4>
<p>We have seen above that, given a curve in the \(xy\) plane, there is no unique way of representing it in parametric form. However, for some commonly occurring curves, particularly the conics, there are accepted standard parametric equations.</p>

<h5> The Parabola </h5>
<p>The standard parametric equations for a parabola are\(\quad\displaystyle x=at^2,\quad y=2at\) </p>
<p> Clearly, we have \(t=\dfrac{y}{2a}\) and by eliminating \(t\) we get \(x=a\left(\dfrac{y^2}{4a^2}\right)\) or \(y^2=4ax\) which we recognise as the standard Cartesian description of a parabola. As an illustration, the below figure shows the curve with \(a=2\) and \(-1\le t\le 2.3\)</p><img src="@@PLUGINFILE@@/17-3-fig19-parabola.svg" alt="A parabola with with a=2 and −1≤t≤2.3" width="300" height="197" class="img-fluid atto_image_button_text-bottom"><br>]]></text>
<file name="17-3-fig19-parabola.svg" path="/" encoding="base64">[encoded data, 14292 characters]</file>
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