1. Tardigrade
  2. Question
  3. Mathematics
  4. The line x=8 is the directrix of the ellipse E: (x2/ a 2)+(y2/b2)=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 √3) and intersects the x-axis at Q, then (3 PQ )2 is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. 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 ____

JEE MainJEE Main 2023Conic Sections

Solution:

$\frac{ a }{ e }=8$....(1)
$\text { ae }=2$....(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)=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$