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10-2. Parabola, Ellipse, Hyperbola
hard
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 $........$
A
$38$
B
$39$
C
$35$
D
$36$
(JEE MAIN-2023)
Solution
$\frac{ a }{ e }=8 \ldots \ldots \ldots(1) \quad ae =2$
$8 e =\frac{2}{ e }$
$e ^2=\frac{1}{4} \Rightarrow e =\frac{1}{2}$
$a =4$
$b ^2= a ^2\left(1- e ^2\right)$
$=16\left(\frac{3}{4}\right) \quad=12$
$\frac{ x \cos \theta}{4}+\frac{ y \sin \theta}{2 \sqrt{3}}=1$
$\sin \theta=\frac{1}{2}$
$\theta=30^{\circ}$
$P (2 \sqrt{3}, \sqrt{3})$
$Q \left(\frac{8}{\sqrt{3}}, 0\right)$
$(3 PQ )^2=39$
Standard 11
Mathematics
