Question

Let tangents be drawn from the point $P(1,3)$ on the ellipse $2x^2+y^2=1$ and points of contact be $A$ and B. The line AB cuts the hyperbola $2x^2-y^2=1$ at $M\&N$ and intersects the parabola $y^2=16x$ 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 $AB$ will be $2x+3y=1$ ....(1)
Homogenizing hyperbola with the help of line, we get
$2x^2-y^2=(2x+3y{)}^2\text{ (i.e. joint equation }OM\text{ and }ON\text{ ) }$
$\Rightarrow 2x^2+12xy+10y^2=0\Rightarrow x^2+6xy+5y^2=0$
$\therefore \tan \theta =\frac{2\sqrt{9-5}}{1+5}=\frac{4}{6}=\frac{2}{3}$
$\Rightarrow \theta ={\tan }^{-1}\left(\frac{2}{3}\right)$