<p>A curve \(C\) has parametric equations</p>
<p style="text-align: center">\[x = t^2 - 1, \quad y = \frac{2}{3}t^3 - 2at \quad \text{for}\hspace{0.5em} t \in \mathbb{R}\]</p>
<p>where&nbsp;\(a\) is a positive constant. \(P\) is a point on \(C\) where&nbsp;\(t = a\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the equation of the tangent to the curve at \(P\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">\(Q\) is another point on \(C\). The tangent to the curve at \(Q\) is parallel to the tangent to the curve at \(P\). Find, in terms of \(a\), the coordinates of \(Q\).<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the range of values of \(a\) for which \(C\) intersects the positive \(x\)-axis.<span class="inline-marks">[2]</span></div></li></ol>

\[x = t^2 - 1, \quad y = \frac{2}{3}t^3 - 2at \quad \text{for } t \in \mathbb{R}\]

where \(a\) is a positive constant. \(P\) is a point on \(C\) where \(t = a\).

Find the equation of the tangent to the curve at \(P\).

\(Q\) is another point on \(C\). The tangent to the curve at \(Q\) is parallel to the tangent to the curve at \(P\). Find, in terms of \(a\), the coordinates of \(Q\).

Find the range of values of \(a\) for which \(C\) intersects the positive \(x\)-axis.

2024 EJC Promo Q9

2024 EJC Promo Q9