<p>The diagram shows an ellipse, centred at the origin with semi-minor axis \(a\) and semi-major axis \(2a\), where&nbsp;\(a &gt; 0\). The rectangle \(PQRS\) is inscribed in the ellipse such that its four vertices \(P\), \(Q\), \(R\) and \(S\) lie on the ellipse. The coordinates of \(P\) is \((x, y)\), where&nbsp;\(x\) and \(y\) are positive.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1779452822217-nm1ayx4ot8n.png" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">Write down the cartesian equation of the ellipse in terms of \(a\).<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">By considering the coordinates of \(P\), show that&nbsp;\(A\), the area of rectangle \(PQRS\), is \(8x\sqrt{a^2 - x^2}\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Using differentiation, find, in terms of \(a\), the value of \(x\) when&nbsp;\(A\) is a maximum. You do not need to prove that&nbsp;\(A\) is a maximum.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the value of \(a\) if the maximum value of \(A\) is \(100\).<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an ellipse, centred at the origin with semi-minor axis \(a\) and semi-major axis \(2a\), where \(a > 0\). The rectangle \(PQRS\) is inscribed in the ellipse such that its four vertices \(P\), \(Q\), \(R\) and \(S\) lie on the ellipse. The coordinates of \(P\) is \((x, y)\), where \(x\) and \(y\) are positive.

Write down the cartesian equation of the ellipse in terms of \(a\).

By considering the coordinates of \(P\), show that \(A\), the area of rectangle \(PQRS\), is \(8x\sqrt{a^2 - x^2}\).

Using differentiation, find, in terms of \(a\), the value of \(x\) when \(A\) is a maximum. You do not need to prove that \(A\) is a maximum.

Find the value of \(a\) if the maximum value of \(A\) is \(100\).

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