Question

From point $P(8,27)$ tangents $PQ$ and $PR$ are draw to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1,$ then angle subtended by $QR$ at origin is

• A. ${\tan }^{-1}\frac{2\sqrt{6}}{65}$
• B. ${\tan }^{-1}\frac{4\sqrt{6}}{65}$
• C. ${\tan }^{-1}\frac{8\sqrt{2}}{65}$
• D. None of these

Answer 1

The correct option is D None of these

Equation of chord of contact $QR$ is
$\frac{8x}{4}+\frac{27y}{9}=1$
$\Rightarrow 2x+3y=1$ ...(i)
Now, equation of the pair of lines passing through origin and points $Q,R$ given by
$(\frac{x^2}{4}+\frac{y^2}{9})=(2x+3y{)}^2$
$\Rightarrow 9x^2+4y^2=36(4x^2+12xy+9y^2)$
$\Rightarrow 135x^2+432xy+320y^2=0$
Required angle is
$={\tan }^{-1}\frac{2\sqrt{(216{)}^2-135\cdot 320}}{455}$
$={\tan }^{-1}\frac{8\sqrt{2916-2700}}{455}$
$={\tan }^{-1}\frac{8\sqrt{216}}{455}$
$={\tan }^{-1}\frac{48\sqrt{6}}{455}$