Let P(4,4√3) be a point on the parabola y^2=4ax and PQ be a focal chord of the parabola.

Question

Let $P(4,4\sqrt{3})$ be a point on the parabola ${y}^{2}=4ax$ and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

Accepted Answer

$\begin{gathered} (4,4\sqrt{3}){y}^{2}=4ax \ \Rightarrow 48=4a.4 \ \, 4a=12 \ \Rightarrow {y}^{2}=12x \ P{t}_{1}=\frac{2}{\sqrt{3}}\Rightarrow Q $

$ {t}_{2}=-\frac{\sqrt{3}}{2}\Rightarrow Q(\frac{9}{4},-3\sqrt{3}) \ \ =\frac{1}{2}MN\cdot (PM+QN) \ =\frac{1}{2}MN\cdot (PS+QS) \ =\frac{1}{2}MN\cdot PQ \ =\frac{1}{2}7\sqrt{3}\cdot \frac{49}{4}=(343)\frac{\sqrt{3}}{8}=3 \end{gathered}$

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