Question

Consider the parabola $y^2 = 4x$. Let $P$ and $Q$ be points on the parabola where $P. (4, -4)$ & $Q (9, 6)$. Let $R$ be a point on the arc of the parabola between $P \& Q$. Then the area of $\Delta $PQR is largest when

• A. $\angle PQR = 90^{\circ}$
• B. $R (4,4)$
• C. $R \left( \frac{1}{4} , 1 \right)$
• D. $R \left( 1, \frac{1}{4} \right)$

Answer 1

Answer: (C)

Equation of $PQ$ is $2x - y = 12$

Perpendicular distance
$L R=\left|\frac{2 t^{2}-2 t-12}{\sqrt{5}}\right|$
$=\frac{2}{\sqrt{5}}(t-3)(t+2)$
$L R$ is Maximum, at $t=\frac{1}{2}$
$\therefore R$ is $\left(\frac{1}{4}, 1\right)$