Show that the locus of the mid-points of all chords passing through the vertices of the parabola `y^(2)`=4ax is the parabola `y^(2)=2ax`.

Explanation

Let AP be a chord of parabola `y^(2)=4ax` which passes through the vertex A and the co-ordinates of point P be `(x_(1),y_(1))`.

Point `P(x_(1),y_(1))` lies on the parabola `y^(2)=4ax`.
`:." "y_(1)^(2)=4ax_(1)` . . .(1)
Let Q be the mid point of AP and its co-ordinates are
(h,k).
`:." "h=(0+x_(1))/(2)andk=(0+y_(1))/(2)`
`rArr" "x_(1)=2handy_(1)=2k`
From eq. (1)
`(2k)^(2)=4a(2h)`
`rArr" "k^(2)=2ah`
`:.`Locus of Q(h, k)
`y^(2)=ax`.