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

Let the chord joining points `P(at_(1)^(2),2at_(1))andQ(at_(2)^(2),2at_(2))` subtends right angle at vertex of the parabola `y^(2)=4ax`
`:." "t_(1)t^(2)=-4`
If midpoint of chord PQ is R (h,k), then
`h=(at_(1)^(2)+at_(2)^(2))/(2),k=(2at_(1)+2at_(2))/(2)`
`:." "(2h)/(a)=(t_(1)+t_(2))^(2)-2t_(1)t_(2)=(t_(1)+t_(2))^(2)+8and(k)/(a)-t_(1)+t_(2)`
Eliminating `t_(1)+t_(2)`, we get
`(2h)/(a)=(k^(2))/(a^(2))+8`
`:." "2ax=y^(2)+8a^(2)`, which is the required locus.