Find the locus of the middle points of the chords of the parabola `y^2=4a x` which subtend a right angle at the vertex of the parabola.

Explanation

We have given equation of parabola `y^2=4a x`

Now let (l,m) be the middle point of chord `t_1t_2`of parabola `y^2=4ax `

`implies 2l=a((t_1)^2+(t_2)^2`

`2m=2a(t_1+t_2)`

`2l=a[(t_1+t_2)^2-2t_1t_2]`

`implies 2l=a[(k^2)/(a^2)+8]`

`implies 2a(h-4a)=k^2`

Hence locus= `y^2=2a(x-2a)`