Question

Find the locus of the middle points of chords of the parabola which pass through the fixed point (h, k).

Answer 1

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

$\therefore s^2-4ar=ys-2ar-2ax$ becomes the equation of chords.

These chords pass through the fixed point $(h,k)$.

Substituting $(h,k)$ in the equation, we have

$s^2-4ar=ks-2ar-2ah$

$\Rightarrow ks+2ar-2ah=s^2$

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