The curve $C$ is defined by the parametric equations $x=at^2$, $y=\frac{a}{t}$, $t \neq 0$,

Question

The curve $C$ is defined by the parametric equations $x=at^2$, $y=\frac{a}{t}$, $t \neq 0$, where $a$ is a positive constant. Show that the equation of the tangent to the curve at the point $P\left(ap^2, \frac{a}{p}\right)$ is $2p^3y+x=3ap^2$.[2] Find the equations of the tangents to $C$ at the points where $x=a$. Find also the acute angle between these tangents.[3] Show that $(q-p)^2(q+2p)=q^3-3p^2q+2p^3$.[1] The tangent to the curve at $P$ cuts the curve again at $Q\left(aq^2, \frac{a}{q}\right)$. Find $q$ in terms of $p$.[3] The following diagram shows the graph of $C$ for $a=1$, and the Cartesian equation of the curve $C$ is $xy^2=1$. The tangent to $C$ at the point $\left(p^2, \frac{1}{p}\right)$ cuts the curve again at the point $\left(4p^2, -\frac{1}{2p}\right)$. The region $S$ is bounded by the tangent to $C$ at the point $\left( p^2, \frac{1}{p} \right)$, the curve $C$ and the line $x = p^2$ Find the area of $S$ in terms of $p$.[3]

Accepted Answer

(a) $2p^3y+x=3ap^2$ (b) $y=-\frac{x}{2}+\frac{3a}{2}$, $y=\frac{x}{2}-\frac{3a}{2}$, acute angle $\tan^{-1}\!\left(\frac{4}{3}\right)\approx53.1^\circ$ (c)(i) $(q-p)^2(q+2p)=q^3-3p^2q+2p^3$ (c)(ii) $q=-2p$ (d) $\frac{11p}{4}$

2025 ACJC P1 Q11

2025 ACJC P1 Q11

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