<p>A curve \(C\) has parametric equations</p>
<p style="text-align: center">\[x = \frac{\lambda}{1 + \lambda^3} \quad \text{and} \quad y = \frac{\lambda^2}{1 + \lambda^3}, \quad \text{for}\hspace{0.5em} \lambda &gt; 0.\]</p>
<p>The point \(P\) is a variable point on \(C\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">With reference to the origin \(O\), \(OP\) forms the diagonal of the rectangle \(OQPR\), where vertices \(Q\) and \(R\) lie on the \(x\)- and \(y\)-axis respectively. Using differentiation, find the value of \(\lambda\) which maximises the area of rectangle \(OQPR\). You need to show that your answer gives a maximum.<span class="inline-marks">[5]</span></div></li></ol>
<p>When the area of rectangle \(OQPR\) is at its maximum, the rate of change of the \(x\)-coordinate of the point \(P\) is \(1\) unit per second.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) and hence determine the rate of change of the \(y\)-coordinate of the point \(P\) at this instant.<span class="inline-marks">[4]</span></div></li></ol>

\[x = \frac{\lambda}{1 + \lambda^3} \quad \text{and} \quad y = \frac{\lambda^2}{1 + \lambda^3}, \quad \text{for } \lambda > 0.\]

The point \(P\) is a variable point on \(C\).

With reference to the origin \(O\), \(OP\) forms the diagonal of the rectangle \(OQPR\), where vertices \(Q\) and \(R\) lie on the \(x\)- and \(y\)-axis respectively. Using differentiation, find the value of \(\lambda\) which maximises the area of rectangle \(OQPR\). You need to show that your answer gives a maximum.

When the area of rectangle \(OQPR\) is at its maximum, the rate of change of the \(x\)-coordinate of the point \(P\) is \(1\) unit per second.

Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) and hence determine the rate of change of the \(y\)-coordinate of the point \(P\) at this instant.

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