### Ellipse Question 5

#### Question 5 - 01 February - Shift 2

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 ________

## Show Answer

#### Answer: 39

#### Solution:

#### Formula: Equation of Tangent ( Parametric form ), Distance Formula

$\frac{a}{e}=8 \ldots \ldots \ldots(1)$

$ ae=2 \ldots \ldots \ldots(2)$

From (1) and (2) ,we get,

$8 e=\frac{2}{e}$

$e^{2}=\frac{1}{4} \Rightarrow e=\frac{1}{2}$

$a=4$

$b^{2}=a^{2}(1-e^{2})$

$=16(\frac{3}{4})=12$

$\frac{x \cos \theta}{4}+\frac{y \sin \theta}{2 \sqrt{3}}=1$

$\sin \theta=\frac{1}{2}$

$\theta=30^{\circ}$

$P(2 \sqrt{3}, \sqrt{3})$

$Q(\frac{8}{\sqrt{3}}, 0)$

$(3 PQ)^{2}=39$