Question

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

Answer 1

Equation of the chord of contact, $2x+3y=1$

Points on the parabola,

$\frac{9(1-3y{)}^2}{4}+4y^2=36$

$97y^2-54y+9=144$

Solving this equation, we get

$y=1.49,-0.934$

Similarly, we get

$x=-1.736,1.901$

Angle substended,

$\theta ={\tan }^{-1}\frac{\frac{-1.49}{1.736}-\frac{-0.934}{1.901}}{1+\frac{1.49}{1.736}\frac{0.934}{1.901}}$

$\theta =\pi -0.2526=2.89$ radians