Question

If a variable tangent of the circle $x^2+y^2=1$ intersects the ellipse $x^2+2y^2=4$ at points $P$ and $Q,$ then the locus of the point of intersection of the tangents at $P$ and $Q$ is

• A. an ellipse with latus rectum $2$ units
• B. a circle of radius $2$ units
• C. a parabola with focus at $(2,3)$
• D. an ellipse with eccentricity $\frac{\sqrt{3}}{4}$

Answer 1

The correct option is A an ellipse with latus rectum $2$ units

Let the point of intersection of the tangents at $P$ and $Q$ to the ellipse $x^2+2y^2=4$ be $(h,k).$
Then, the equation of $PQ$ is $T=0$
$\Rightarrow hx+2ky=4$
Since this line is tangent to the circle $x^2+y^2=1$
$\therefore \left|\frac{0+0-4}{\sqrt{h^2+4k^2}}\right|=1$
$\Rightarrow h^2+4k^2=16$
$\therefore$ Required locus is
$\frac{x^2}{16}+\frac{y^2}{4}=1$
which is equation of an ellipse.
$\therefore$ Lenght of latus rectum$=\frac{2b^2}{a}=\frac{8}{4}=2$ units
and eccentricity $=\sqrt{1-\frac{4}{16}}=\frac{\sqrt{3}}{2}$