The locus of the mid-point of the line segment joining a point on the parabola `Y^(2)=4ax` and the point of contact of circle drawn on focal distance of the point as diameter with the tangent at the vertex, is

Options

A. `y^(2)=9ax`
B. `9y^(2)=2ax`
C. `2x^(2)=9ay`
D. `2y^(2)=9ax`

Answer

D

Explanation

Let `P(at^(2), 2at)` be a point on the parabola. The equation of the circle drawn on the focal distance SP as diameter is
`(x-a)(x-at^(2))+(y-0)(y-2at)=0`
The tangent at the vertex i.e. x=0 cuts the circle at Q(0, at). Let R(h, k) be the mid-point PQ. Then,
`2h=at^(2)" and "2k=3atrArr2h=a((2k)/(3a))^(2)rArr2k^(2)=9ah`
Hence, the locus of (h, k) is `2y^(2)=9ax`.