Equation of the ellipse whose focus is $(6, 7)$ directrix is $x + y + 2 = 0$ and $e = \frac{1}{\sqrt{3}}$ is:

5 x^{2} + 2 x y + 5 y^{2} - 76 x - 88 y + 506 = 0

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5 x^{2} - 2 x y + 5 y^{2} - 76 x - 88 y + 506 = 0

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5 x^{2} - 2 x y + 5 y^{2} + 76 x + 88 y - 506 = 0

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Solution

The correct option is B $5 x^{2} - 2 x y + 5 y^{2} - 76 x - 88 y + 506 = 0$
Focus $: (6, 7)$ , directrix $: x + y + 2 = 0$ , $e = \frac{1}{\sqrt{3}}$
equation of ellipse: $(x - 6)^{2} + (y - 7)^{2} = (\frac{1}{\sqrt{3}})^{2} (\frac{x + y + 2}{\sqrt{1 + 1}})^{2}$
$\Rightarrow 6 (x^{2} + y^{2} + 36 + 49 - 12 x - 14 y) = x^{2} + y^{2} + 4 + 2 x y + 4 x + 4 y$
$\Rightarrow 5 x^{2} + 5 y^{2} - 2 x y - 76 x - 88 y + 506 = 0$

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