Question

The straight line joining any point $P$ on the parabola $y^2=4ax$ to the vertex and perpendicular from the focus to the tangent at $P,$ intersect at $R,$ then the equation of the locus of $R$ is

• A. $x^2+2y^2-ax=0$
• B. $2x^2+y^2-2ax=0$
• C. $2x^2+2y^2-ay=0$
• D. $2x^2+y^2-2ay=0$

Answer 1

Answer: (B)

$2x^2+y^2-2ax=0$

Given the equation of parabola is
$y^2=4ax$

Let $P(a{t}^2,2at)$ be any point on the parabola.

Equation of tangent at point P is $ty=x+a{t}^2$ where slope of the tangent is $\frac{1}{t}.$

Equation of line perpendicular to the tangent passes through $(a,0)$ is given as
$\therefore y-0=-t(x-a)$

or $y=t(a-x)$ .....(i)

Equation of $OP$ is given by

$y-0=\frac{2}{t}(x-0)=0$

$\Rightarrow y=\frac{2}{t}x$ .....(ii)

Eliminating 't' from equations (i) and (ii), we get

$y^2=2x(a-x)$

or $2x^2+y^2-2ax=0$