A curve $C$ has parametric equations $$x=4t^2+1,\qquad y=2t.$$ The points $P$ and $Q$ on $C$

Question

A curve $C$ has parametric equations $$x=4t^2+1,\qquad y=2t.$$ The points $P$ and $Q$ on $C$ have parameters $p$ and $3p$ respectively. The point $M$ is the midpoint of $PQ$. Show that the coordinates of $M$ are $(20p^2+1,4p)$, and hence find the Cartesian equation of the locus of $M$ as $p$ varies.[4] The line $L$ has equation $$x=15y-39.$$ It intersects the locus of $M$ at two points. Find the corresponding values of $p$.[3] For the smaller value of $p$, denote the corresponding points on $C$ by $P_1$ and $Q_1$. For the larger value, denote them by $P_2$ and $Q_2$. The four distinct points $P_1,Q_1,P_2,Q_2$ form a quadrilateral. Find the exact area of the quadrilateral. [5]

Accepted Answer

(a) $x=\dfrac54y^2+1$ (b) $p=1\text{ or }p=2$ (c) $128\text{ units}^2$

Area from a parametric midpoint locus

Area from a parametric midpoint locus

Video Solution:

Solution:

Solution locked