Question

A tangent to the ellipse $x^2+4y^2=4$ meets the ellipse $x^2+2y^2=6$ at $P$ and $Q$. Then the angle between pair of tangents at $P$ and $Q$ of the ellipse $x^2+2y^2=6$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is C $\frac{\pi }{2}$

Let $R(h,k)$ be the point of intersection of tangents at $P$ and $Q$ of the ellipse $x^2+2y^2=6$. $PQ$ is the chord of contact of $x^2+2y^2=6$ w.r.t. the point $R(h,k)$
$T=0\Rightarrow hx+2ky-6=\mathrm{0.....}\left(1\right)$
This line is tangent to the ellipse $x^2+4y^2=4$
Let $(2\cos \theta ,\sin \theta )$ be any point on the ellipse $x^2+4y^2=4.$ The tangent at this point is given by $2\cos \theta x+4\sin \theta y-4=\mathrm{0.....}\left(2\right)$

Comparing $\left(1\right)$ and $\left(2\right)$
$\Rightarrow \frac{h}{2\cos \theta }=\frac{k}{4\sin \theta }=\frac{-6}{-4}$
$\Rightarrow h=3\cos \theta ,k=3\sin \theta$
$\Rightarrow h^2+k^2=9$

$(h,k)$ lies on the director circle.