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

(b)$y=\frac{1}{2}x-\frac{3a}{2}$ and angle =$\,\mathrm{53.1}^\circ$ or 0.927 radians (c)(i)$q^3-3p^2q+2p^3$ (c)(ii)$q=-2p$ (d)$\frac{11p}{4}$

2025 ACJC P1 Q11

2025 ACJC P1 Q11