Question

Through the vertex $O$ of a parabola $y^2 = 4x$, chords $OP$ and $OQ$ are drawn at right angles to one another. The locus of the middle point of $PQ$ is

• A. $y^2 = 2x + 8$
• B. $y^2 = x + 8$
• C. $y^2 = 2x - 8$
• D. $y^2 = x - 8$

Answer 1

Answer: (C)

$y^2 = 2x - 8$

Answer 2

Given parabola is $y^2 = 4x$ ....(1) Let $P \equiv\left(t^{2}_{1} , 2t_{1}\right)$ and $Q \equiv \left(t^{2}_{2}, 2t_{2}\right)$ Slope of $OP = \frac{2t_{1}}{t_{1}^{2}} = \frac{2}{t_{1}}$ and slope of $OQ = \frac{2}{t_{2}}$ Since $OP \bot OQ$, $\therefore \frac{4}{t_{1}t_{2}} = - 1 t_{1 }t_{2} = - 4$ ....(2) Let $R (h, k)$ be the middle point of $PQ$, then $h = \frac{t_{1}^{2} + t^{2}_{2}}{ 2}$ ....(3) and $k = t_{1} +t_{2}$ ....(4) From (4) , $k^{2} = t^{2}_{1} + t^{2}_{2} + 2t_{1}t_{2} = 2h-8$ [ From (2) and (3)] Hence locus of $R (h, k)$ is $y^2 = 2x - 8$.