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 :
$${y^2} = 2x - 8$$
Solution :
Given parabola is $${y^2} = 4x......\left( 1 \right)$$
Let $$P \equiv \left( {t_1^2,\,2{t_1}} \right){\text{ and }}Q \equiv \left( {t_2^2,\,2{t_2}} \right)$$
Slope of $$OP = \frac{{2{t_1}}}{{t_1^2}} = \frac{2}{{{t_1}}}$$ and slope of $$OQ = \frac{2}{{{t_2}}}$$
Since $$OP \bot OQ,$$
$$\therefore \,\frac{4}{{{t_1}{t_2}}} = - 1{\text{ or }}{t_1}{t_2} = - 4......\left( 2 \right)$$
Let $$R\left( {h,\,k} \right)$$ be the middle point of $$PQ,$$ then
$$h = \frac{{t_1^2 + t_2^2}}{2}......\left( 3 \right)$$
and $$k = {t_1} + {t_2}......\left( 4 \right)$$
From $$\left( 4 \right),\,{k^2} = t_1^2 + t_2^2 + 2{t_1}{t_2} = 2h - 8\,\,\,\,\left[ {{\text{From}}\left( 2 \right){\text{and}}\left( 3 \right)} \right]$$
Hence locus of $$R\left( {h,\,k} \right)$$ is $${y^2} = 2x - 8.$$