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  • The line x = 8 is the directrix of the ellipse ellipse <img alt="E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1" src="https://entrancecorner.oncodecogs.com/gif.latex?E%3A%20%5Cfrac%7Bx%5E2%7D%7Ba%5E2%7D

The line x = 8 is the directrix of the ellipse 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 that x-axis at Q then (3 P Q)^2 equl to

Option: 1

39


Option: 2

__


Option: 3

__


Option: 4

__


Answers (1)

best_answer

\begin{aligned} & \frac{\mathrm{a}}{\mathrm{e}}=8 \\ & \mathrm{ae}=2 \\ & 8 \mathrm{e}=\frac{2}{\mathrm{e}} \\ & \mathrm{e}^2=\frac{1}{4} \Rightarrow \mathrm{e}=\frac{1}{2} \\ & \mathrm{a}=4 \\ & \mathrm{~b}^2=\mathrm{a}^2\left(1-\mathrm{e}^2\right) \\ & =16\left(\frac{3}{4}\right)=12 \end{aligned}

\begin{aligned} & \frac{x \cos \theta}{4}+\frac{y \sin \theta}{2 \sqrt{3}}=1 \\ & \sin \theta=\frac{1}{2} \\ & \theta=30^{\circ} \end{aligned}

\begin{aligned} & P(2 \sqrt{3}, \sqrt{3}) \\ & Q\left(\frac{8}{\sqrt{3}}, 0\right) \\ & (3 P Q)^2=39 \end{aligned}

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