Question

Let tangents be drawn from the point $P (1,3)$ on the ellipse $2 x ^2+ y ^2=1$ and points of contact be $A$ and B. The line AB cuts the hyperbola $2 x^2-y^2=1$ at $M \& N$ and intersects the parabola $y^2=16 x$ at $Q$ and $R$.
The angle subtended by $MN$ at the centre of hyperbola will be

• A. $\frac{\pi}{2}$
• B. $\tan ^{-1}\left(\frac{2}{3}\right)$
• C. $\tan ^{-1}\left(\frac{3}{2}\right)$
• D. $\tan ^{-1} 2$

Answer 1

Answer: (B)

Clearly, AB is chord of contact
$\therefore$ Equation of $A B$ will be $2 x+3 y=1$ ....(1)
Homogenizing hyperbola with the help of line, we get
$2 x^2-y^2=(2 x+3 y)^2 \text { (i.e. joint equation } O M \text { and } O N \text { ) } $
$\Rightarrow 2 x^2+12 x y+10 y^2=0 \Rightarrow x^2+6 x y+5 y^2=0 $
$\therefore \tan \theta=\frac{2 \sqrt{9-5}}{1+5}=\frac{4}{6}=\frac{2}{3} $
$\Rightarrow \theta=\tan ^{-1}\left(\frac{2}{3}\right) $