Let $P(4,4\sqrt3)$ be a point on the parabola $y^2=4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the feet of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral $PQMN$ is equal to :

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $48=16a\Rightarrow a=3$; $P$ has parameter $t=\frac2{\sqrt3}$, so $Q$ has parameter $-\frac1t=-\frac{\sqrt3}2$, giving $Q\left(\frac94,-3\sqrt3\right)$. The directrix is $x=-3$, so $PM=7$ and $QN=\frac{21}4$ are parallel, with separation $4\sqrt3+3\sqrt3=7\sqrt3$. Area of the trapezium $=\frac12\left(7+\frac{21}4\right)7\sqrt3=\frac{343\sqrt3}8$. Answer (4).

Q. Let $P(4,4\sqrt3)$ be a point on the parabola $y^2=4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the feet of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral $PQMN$ is equal to :

Explanation

Reviewed by Dr. Abubacker

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