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

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

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

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

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

answer is A.

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Detailed Solution

Common tangents to $x^{2} + y^{2} = 1 a n d x^{2} - y^{2} = 1.$ are $x = \pm 1$

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