Question

If tangent to parabola $y^2=4ax$ intersects ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at $A$ and $B,$ then the locus of point of intersection of tangents at $A$ and $B,$ is

• A. straight line
• B. parabola with length of latus rectum $\frac{b^4}{a^3}$ unit
• C. ellipse
• D. parabola with length of latus rectum $\frac{a^4}{b^3}$ unit

Answer 1

The correct option is B parabola with length of latus rectum $\frac{b^4}{a^3}$ unit

Let $P(h,k)$ be the point of intersection of tangents at $A$ and $B.$
So, equation of chord of contact $AB$ is
$\frac{hx}{a^2}+\frac{ky}{b^2}=1\dots \left(i\right)$

Also, equation of tangent to parabola $y^2=4ax$ is
$y=mx+\frac{a}{m}\Rightarrow mx-y=\frac{-a}{m}\dots \left(ii\right)$

If equation $\left(i\right)$ and $\left(ii\right)$ are identical, then
$\frac{m}{h/a^2}=\frac{-1}{k/b^2}=\frac{-a/m}{1}$
$\Rightarrow m=\frac{-h{b}^2}{k{a}^2}$ and $m=\frac{ak}{b^2}$
$\Rightarrow \frac{-h{b}^2}{k{a}^2}=\frac{ak}{b^2}$

Hence, required locus is
$y^2=\frac{-b^4}{a^3}x$
which is a parabola, with length of latus rectum $\frac{b^4}{a^3}$ units.