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<quiz>
  <question type="stack">
    <name>
      <text>17.3.1.4 Parametric Curves Task</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<p>Consider the following two parametric representations of curves:</p>
<ol>
<li>\(\quad x=\cos t,\qquad y=\sin t,\qquad 0\le t\le \dfrac{\pi}{2}.\)</li>
<li>\(\quad x=t,\qquad\quad y=\sqrt{1-t^2},\qquad 0\le t\le 1.\)</li>
</ol>

<ol class="HELM_parts">
<li><p>Eliminate the parameter \(t\) from the parametric equations in 1., writing an equation in terms of \(x\) and \(y\): [[input:ans1]] [[validation:ans1]] </p> 
<p>[Hint: what happens if you square both expressions for \(x\) and \(y\)?]</p>
<p>[[feedback:prt1]]</li>
<li><p>Eliminate the parameter \(t\) from the parametric equations in 2., writing an equation in terms of \(x\) and \(y\): [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p></li>
<li><p>What do you conclude about the 2 parametric representations? [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p></li>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text><![CDATA[<p> We have the 2 parametric representations:</p>
<ol>
<li>\(\quad x=\cos t,\qquad y=\sin t,\qquad 0\le t\le \dfrac{\pi}{2}.\)</li>
<li>\(\quad x=t,\qquad\quad y=\sqrt{1-t^2},\qquad 0\le t\le 1.\)</li>
</ol>

<p> (a) We are asked to eliminate the parameter \(t\) from the parametric equations in 1., writing an equation in terms of \(x\) and \(y\).</p>
<p> To do this, we square both of our expressions:</p>
<p> \(\qquad x = \cos t \implies x^2 = \cos^2 t, \quad \text{and} \quad y= \sin t \implies y^2 = \sin^2 t,\) so</p>
<p> \( \qquad x^2 + y^2 = \cos^2 t + \sin^2 t = 1.\)</p><br>

<p> (b) We are asked to eliminate the parameter \(t\) from the parametric equations in 2., writing an equation in terms of \(x\) and \(y\).</p>
<p> This is similar. We have:</p>
<p> \(\qquad x=t \implies y = \sqrt{1-x^2} \implies y^2 = 1 - x^2 \implies  x^2 + y^2 =1.\)</p><br>

<p> (c) Based on the above answers, we can see that both parametric descriptions represent the same curve - (part of) a circle centred at the origin of radius 1. </p>]]></text>
    </generalfeedback>
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    <idnumber/>
    <stackversion>
      <text>2025040100</text>
    </stackversion>
    <questionvariables>
      <text><![CDATA[ta1:x^2 + y^2 = 1;
alt1: y = sqrt(1-x^2);
ta2:x^2 + y^2 = 1;
alt2: y = sqrt(1-x^2);
ta3:[["They describe different curves",false],["They describe the same curve",true],["They describe different parts of the same curve",false], ["They describe the same curve, but in different directions", false]];]]></text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
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    <questionnote format="html">
      <text>\(x,y\) equation: {@ta1@}</text>
    </questionnote>
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      <text/>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
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    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
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      <description>Test case assuming the teacher's input gets full marks.</description>
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  </question>
</quiz>
