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An ellipse has eccentricity $\frac{1}{2}$ and one focus at $S (\frac{1}{2}, 1) .$ Its one directrix is common tangent (nearer to $s$ ) to the circle $x^{2} + y^{2} = 1$ and $x^{2} - y^{2} = 1 .$ The equation of the ellipse is:

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$9 {(x - \frac{1}{3})}^{2} + 12 {(y - 1)}^{2} = 1$

$12 {(x - \frac{1}{3})}^{2} + 9 {(y - 1)}^{2} = 1$

$3 {(x - \frac{1}{3})}^{2} + 4 {(y - 1)}^{2} = 1$

$9 {(x - \frac{1}{3})}^{2} + 16 {(y - 1)}^{2} = 1$

answer is A.

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Detailed Solution

Directrix $\Rightarrow x = 1$

focus $= (\frac{1}{2}, 1)$

$e = \frac{1}{2}$

$∵ \frac{P S}{P M} = e \Rightarrow P S^{2} = e^{2} P M^{2} \Rightarrow {(x - \frac{1}{2})}^{2} + {(y - 1)}^{2} = {(\frac{1}{2})}^{2} {(x - 1)}^{2} \Rightarrow 9 {(x - \frac{1}{3})}^{2} + 12 {(y - 1)}^{2} = 1$

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