Question

The line $x=8$ is the directrix of the ellipse $E : \frac{x^2}{ a ^2}+\frac{y^2}{b^2}=1$ with the corresponding focus $(2,0)$ If the tangent to $E$ at the point $P$ in the first quadrant passes through the point $(0,4 \sqrt{3})$ and intersects the $x$-axis at $Q$, then $(3 PQ )^2$ is equal to ____

Answer 1

The correct answer is 39
ea​=8....(1)
ae =2....(2)
8e=e2​
e2=41​⇒e=21​
a=4
b2=a2(1−e2)
=16(43​)=12
4xcosθ​+23​ysinθ​=1
sinθ=21​
θ=30∘
P(23​,3​)
Q(3​8​,0)
(3PQ)2=39