Question

If the locus of mid point of any normal chord of the parabola $y^2=4x$ is $x-a=\frac{b}{y^2}+\frac{y^2}{c};$ where $a,b,c\in N$, then $(a+b+c)$ is equal to

Answer 1

Let the mid point of the normal chord be $(h,k)$
Then the equation of the chord is
$T=S_1\Rightarrow yk-2(x+h)=k^2-4h$

If the chord is normal to the parabola at $P(t^2,2t)$, then
$t=-\frac{2}{k}\therefore P=(\frac{4}{k^2},-\frac{4}{k})$
$\because P$ also lies on the chord
$\therefore -4-\frac{8}{k^2}-2h=k^2-4h\Rightarrow h-2=\frac{4}{k^2}+\frac{k^2}{2}$
Hence locus is
$\Rightarrow x-2=\frac{4}{y^2}+\frac{y^2}{2}$

On comparing with
$x-a=\frac{b}{y^2}+\frac{y^2}{c}$
We get,
$a=2,b=4,c=2\therefore a+b+c=8$