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<quiz>
  <question type="stack">
    <name>
      <text>6.3.12 (Parametric ellipse)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Find a parametric representation \(x=f(t), y = g(t)\), of the equation \(\frac{x^2}{{@a@}^2} + \frac{y^2}{{@b@}^2} = {@r@}^2\).&nbsp; Your answers should be expressed in terms of a variable \(t\).&nbsp; <br></p><p>(Hint: if \(f(t)\) and \(g(t)\) satisfy the equation, what do the functions \(f(t)/{@a@}\) and \(g(t)/{@b@}\) satisfy?)</p><p>\(x(t) = \) [[input:ans1]]; [[validation:ans1]]</p><p>\(y(t) = \) [[input:ans2]]. [[validation:ans2]]<br></p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>The functions \(f(t) = {@x@}\) and \(g(t) = {@y@}\) satisfy \begin{align*}\frac{\left({@x@} \right)^2}{{@a@}^2} + \frac{\left({@y@}\right)^2}{{@b@}^2} &amp; = \frac{{@r@}^2{@a@}^2 \cos^2(t)}{{@a@}^2} + \frac{{@r@}^2{@b@}^2 \sin^2(t)}{{@b@}^2}\\ &amp; = {@r@}^2\left(\cos^2(t) + \sin^2(t)\right) \\ &amp;= {@r@}^2,\end{align*}so these are parametric representations of the equation. If you decide to plot this function, you would see that this equation is actually the equation of an ellipse.<br></p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020081700</text>
    </stackversion>
    <questionvariables>
      <text>p:rand_range(-10,10,1);
q:rand_range(-10,10,1);
a:2+rand(4);
b:rand_with_prohib(2,5,[a]);
r:rand_with_prohib(2,10,[p,q]);
x: a*r*cos(t);
xda:r*cos(t);
y:b*r*sin(t);
ydb:r*sin(t);
ta1:x
ta2:y</text>
    </questionvariables>
    <specificfeedback format="html">
      <text>[[feedback:prt1]]</text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(\frac{x^2}{{@a@}^2} + \frac{y^2}{{@b@}^2} = {@r@}^2\) has \(x(t) = {@x@}\) amd \(y(t) = {@y@}\).</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
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    <prtcorrect format="html">
      <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>
    </prtpartiallycorrect>
    <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>
    <scientificnotation>*10</scientificnotation>
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    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
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      <name>ans1</name>
      <type>algebraic</type>
      <tans>ta1</tans>
      <boxsize>15</boxsize>
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      <syntaxhint/>
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      <showvalidation>3</showvalidation>
      <options/>
    </input>
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      <name>ans2</name>
      <type>algebraic</type>
      <tans>ta2</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
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      <checkanswertype>0</checkanswertype>
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      <options/>
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      <name>prt1</name>
      <value>1.0000000</value>
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      <feedbackvariables>
        <text>dist:ans1^2/a^2 + ans2^2/b^2</text>
      </feedbackvariables>
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        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>dist</sans>
        <tans>r^2</tans>
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        <trueanswernote>prt1-1-T</trueanswernote>
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        <falsescoremode>=</falsescoremode>
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    <qtest>
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      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2</value>
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      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
