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

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JEE MAIN Mathematics QUESTION #970

Question 1

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

\(17\sqrt{3}\)
\(\dfrac{263\sqrt{3}}{8}\)
\(\dfrac{34\sqrt{3}}{3}\)
\(\dfrac{343\sqrt{3}}{8}\)✔️

Correct Answer Explanation

From \(P(4,4\sqrt{3})\) on \(y^2=4ax\): \((4\sqrt{3})^2=4a\cdot4 \Rightarrow a=3\). So directrix is \(x=-3\). Using focal chord properties, find \(Q\), then compute the trapezium area \(PQMN = \frac{1}{2}(PM+QN)\cdot PQ_x\)-component, giving \(\frac{343\sqrt{3}}{8}\).

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