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The equation of an ellipse whose focus is (-1, 1 ), whose directrix is x - y + 3 = 0 and whose eccentricity is $\frac{1}{2}$ is
given by:

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$7 x^{2} + 2 x y + 7 y^{2} + 10 x - 10 y + 7 = 0$

$7 x^{2} - 2 x y + 7 y^{2} - 10 x + 10 y + 7 = 0$

$7 x^{2} - 2 x y + 7 y^{2} - 10 x - 10 y - 7 = 0$

$7 x^{2} - 2 x y + 7 y^{2} + 10 x + 10 y - 7 = 0$

answer is A.

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Detailed Solution

Let any point on it be (x, y) then by definition,

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

Squaring and simplifying, we get

$7 x^{2} + 2 x y + 7 y^{2} + 10 x - 10 y + 7 = 0$

which is the required ellipse.

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