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 $(3PQ{)}^2$ is equal to ____

Answer 1

Answer: 39

$\frac{a}{e}=8$....(1)
$\text{ ae }=2$....(2)
$8e=\frac{2}{e}$
$e^2=\frac{1}{4}\Rightarrow e=\frac{1}{2}$
$a=4$
$b^2=a^2\left(1-e^2\right)$
$=16\left(\frac{3}{4}\right)=12$
$\frac{x\cos \theta }{4}+\frac{y\sin \theta }{2\sqrt{3}}=1$
$\sin \theta =\frac{1}{2}$
$\theta =30^{\circ }$
$P(2\sqrt{3},\sqrt{3})$
$Q\left(\frac{8}{\sqrt{3}},0\right)$
$(3PQ{)}^2=39$