Question

If the locus of mid point of the chords of the parabola $y^2=4ax$ which passes through a fixed point $(h,k)$ is also a parabola, then length of its latus rectum (in units) is

• A. $4a$
• B. $8a$
• C. $6a$
• D. $2a$

Answer 1

The correct option is D $2a$

Equation of chord having $P(x_1,y_1)$ as mid point is
$T=S_1$
$\Rightarrow y{y}_1-2a(x+x_1)=y_1^2-4a{x}_1$
It passes through $(h,k)$
$\Rightarrow k{y}_1-2a(h+x_1)=y_1^2-4a{x}_1$
Locus of $P$ is
$y^2-ky-2ax+2ah=0\Rightarrow {(y-\frac{k}{2})}^2=2a(x+\frac{k^2}{8a}-h)$
$\therefore$ Length of latus rectum $=2a$ units