Question

Find the locus of the middle points of chords of the parabola which are normal to the curve.

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 are normal to the parabola and so must be of the form $y+tx=2at+a{t}^3$.

Comparing the two equations, we have

$\frac{k}{1}=\frac{-2a}{t}=\frac{k^2-2ah}{2at+a{t}^3}$

$\Rightarrow t=\frac{-2a}{k}$

Also, $2at+a{t}^3=\frac{k^2-2ah}{k}$

$\Rightarrow \frac{-4a^2}{k}+\frac{-8a^4}{k^3}=\frac{k^2-2ah}{k}$

i.e. $-4a^2{k}^2-8a^4=k^4-2ah{k}^2$

i.e. $y^4+y^2(4a^2-2ax)+8a^4=0$ becomes the required locus.