Question

The locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the parabola $y^2=4bx$ is

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

Answer 1

The correct option is B a parabola

The equation of any tangent to $y^2=4ax$ is

$y=mx+\frac{a}{m}$ ...(i)

Let $(h,k)$ be the pole of (i) with respect to the parabola $y^2=4bx$.

Then, the equation of the polar is

$ky=2b(x+h)$

$\Rightarrow 2bx-ky+2bh=0$ ....(ii)

Clearly, (i) and (ii) represent the same straight line.

$\therefore \frac{2b}{m}=\frac{-k}{-1}=\frac{2bh}{a/m}$

$\Rightarrow k=\frac{2b}{m}$ and $h=\frac{a}{m^2}$

$\Rightarrow m=\frac{2b}{k}$ and $m^2=\frac{a}{h}$

$\Rightarrow {\left(\frac{2b}{k}\right)}^2=\frac{a}{h}⟹k^2=\frac{4b^{2}h}{a}$

Hence, the locus of $(h,k)$ is $y^2=\frac{4b^{2}x}{a}$, which is a parabola.