The line x = 8 is the directrix of the ellipse E: x ^ 2 a ^ 2 + y ^...

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The line $x = 8$ is the directrix of the ellipse $E: \dfrac{x2}{a2} + \dfrac{y2}{b2} = 1$ with the corresponding focus \left(2, 0\right)$,</p><p>So, equation of directrix is given by$x = \dfrac{a}{e} ⇒ 8 = \dfrac{a}{e} \ldots \left(1\right)$</p><p>And distance of corresponding focus is given by,</p><p>$ae = 2 \ldots \left(2\right)$</p><p>So, from both equation we get,</p><p>$8e = \dfrac{2}{e}$</p><p>$⇒ e^{2} = \dfrac{1}{4}⇒ e = \dfrac{1}{2}$</p><p>Hence,$a = 4$</p><p>Now using eccentricity formula we get,</p><p>$b^{2} = a^{2}\left(1 - e^{2}\right)= 16\left(\dfrac{3}{4}\right) =12$</p><p>Now equation of tangent to ellipse at point$P \l$left(4cos\theta, \s$ sqrt{12}sin\theta)$is given by,</p><p>$\dfrac{xcos\theta}{4} +\dfrac{ysin\theta}{2\sqrt{3}} =1$</p><p>Now given tangent passes through$\left(0, 4\sqrt{3}\right),

So, sin\theta = \dfrac{1}{2}$⇒$\theta =30 °$</p><p>Now, point$P \$lleft(4cos\theta, \s$ ssqrt{12}sin\theta)will be&P \l left(2\sqrt{3}, \sqrt{3})$</p><p>And intersection of tangent with$x -$axis will give$Q $\left(\dfrac{8}{3},0\right)$

Hence, by distance formula we get, \left(3PQ\right)2 = 39$