Question

Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2=2ax$ is the circle $x^2+y^2=ax$.

Answer 1

Tangent to the parabola $y^2=4ax$ is $ty=x+a{t}^2$

For the circle $x^2+y^2=2ax$, applying $T=0$ gives $xh+yk=ax+ah$ where $(h,k)$ is the pole of the tangent.

Comparing this equation with the tangent equation, we get

$\frac{t}{k}=\frac{1}{a-h}=\frac{a{t}^2}{ah}$

$\Rightarrow t=\frac{k}{a-h},t^2=\frac{h}{a-h}$

$\therefore \frac{h}{a-h}={\left(\frac{k}{a-h}\right)}^2$

$\therefore h(a-h)=k^2$

i.e. $ah-h^2=k^2$

i.e. $x^2+y^2=ax$ is the required locus.