<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>6.3.8 (Parametric form of a circle)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h1>Parametric form of a circle</h1>

<p>We have seen in the previous example that we can parameterise a circle by using the trigonometric functions \(\sin \theta\) and \(\cos \theta\). This section will focus on using parametric representations of circles and identifying when a set of parametric equations gives us a circle. Let us begin by considering the following example.</p>

<h3>Example</h3>

<p>Find a parametric representation \(x = f(t), y = g(t)\) of the circle of radius \(9\) and centre \((-8,-8)\).</p>

<h3>Solution</h3>

<p>In this case, we know that the \(x\) coordinate will be a function of \(\cos t\) and the \(y\)-coordinate will be a function of \(\sin t\). Recall that we can write the Cartesian equation of this circle is given in centre-radius form as 
\[
(x+8)^2 + (y+8)^2 = 9^2.
\]
Then, notice that the functions \(f(t) = 9 \cos t -8\) and \(g(t) = 9 \sin t - 8\) satisfy:
\[
\begin{align*}
(f(t) - (-8))^2 + (g(t) - (-8))^2 &= (9 \cos t)^2 + (9 \sin t)^2
\\
&= 9^2 \cos^2 t + 9^2 \sin^2 t
\\
&= 9^2 (\cos^2 t + \sin^2 t)
\\
&= 9^2. 
\end{align*}
\]
Therefore a parametric representation of the circle is given by \(x = 9 \cos t - 8\), \(y = 9 \sin t -8\).
</p>
<br/>
In general for a circle with centre-radius form \((x-a)^2 + (y-b)^2 = r^2\) the parametric form of the circle is given by
\[
x = r \cos t + a \qquad \mbox{ and } \qquad y = r \sin t + b.
\]]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
