A curve $C$ has parametric equations $$x=t^2+1,\qquad y=t^3-2.$$ The points $P$ and $Q$ on \(C

Question

A curve $C$ has parametric equations $$x=t^2+1,\qquad y=t^3-2.$$ The points $P$ and $Q$ on $C$ have parameters $p$ and $2p$ respectively, where $p>0$. The tangents to $C$ at $P$ and $Q$ meet at $R$. Show that the tangent to $C$ at the point with parameter $t$ has equation $$3tx-2y=t^3+3t+4.$$[3] Find the coordinates of $R$ in terms of $p$.[4] Show that the area of triangle $PQR$ is $\dfrac{5}{3}p^5$. [3] Given that the area of triangle $PQR$ is $\dfrac{160}{3}$, find the value of $p$.[2]

Accepted Answer

(a) $3tx-2y=t^3+3t+4$ (b) $R\left(1+\dfrac73p^2,\ 3p^3-2\right)$ (c) $\dfrac53p^5$ (d) $p=2$

Area enclosed by two parametric tangents

Area enclosed by two parametric tangents

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