The locus of pole of tangents to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` with respect to the parabola `y^(2)=4ax,` is

Options

A. `b^(2)y^(2)=a^(2)(x^(2)-a^(2))`
B. `b^(2)y^(2)=4a^(2)(x^(2)+a^(2))`
C. `b^(2)y^(2)=4a^(2)(x^(2)-a^(2))`
D. `b^(2)y^(2)=4b^(2)(x^(2)-a^(2))`

Answer

C

Explanation

Let (h,k) be the pole of a tangent to the ellipse with respect to the parabola `y^(2)=4ax` then , the equation of the polar is
`ky=2a(x+h) or , y=((2a)/(k))x+(2ah)/(k)`
this touches the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1.`
` therefore ((2ah)/(k))^(2)=A^(2)((2a)/(k))^(2)+b^(2)implies b^(2) k^(2)=4a^(2)h^(2)-4a^(4)`
hence ,the locus of `(h,k)`is `b^(2)y^(2)=4a^(2)(x^(2)-a^(2))`