Question

Find the locus of the middle points of chords of the parabola which pass through the focus.

Answer 1

Equation of a chord with midpoint $(h,k)$ is given by $T=S(h,k)$

$\therefore k^2-4ah=yk-2ah-2ax$ becomes the equation of chords.

These chords pass through the focus.

Substituting $(a,0)$ in the equation, we have

$k^2-4ah=0-2ah-2a^2$

$\Rightarrow 2ah-2a^2=k^2$

i.e. $y^2=2a(x-a)$ becomes the required locus.