The locus of the poles of tangents to the parabola `y^(2)=4ax` with respect to the parabola `y^(2)=4ax` is

Options

A. a circle
B. a parabola
C. a straight line
D. an ellipse

Answer

B

Explanation

The equation of any tangent to `y^(2)=4ax` is
`y=mx+a/m" ….(i)"`
Let (h, k) be the ploe of (i) with respect to the parabola `y^(2)=4ax`.
`ky=2b(x+y)`
`"or, "2bx-ky+2ky=0" ....(ii)"`
Clearly, (i) and (ii) represent the same straight line.
`:." "(2b)/m=(-k)/(-1)=(2bh)/("a/m")`
`rArr" "k=(2b)/m" and "m^(2)=a/h`
`rArr" "m=(2b)/k" and "m^(2)=a/h`
`rArr" "((2b)/k)^(2)=a/hrArrk^(2)=(4b^(2)h)/a`
Hence, the locus of (h, k) is `y^(2)=(4b^(2)x)/a`, which is a parabola.