Find the equation of the ellipse whose focus is the point $(6, 7),$ directrix $x + y + 2 = 0$ and eccentricity $\frac{1}{\sqrt{3}} .$

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

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

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

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

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Solution

The correct option is C $5 x^{2} + 5 y^{2} - 76 x - 88 y - 2 x y + 506 = 0$
By definition $S P = e . P M$
$\Rightarrow \sqrt{{{(x - 6)}^{2} + {(y - 7)}^{2}}} = \frac{1}{\sqrt{3}} | \frac{x + y + 2}{\sqrt{(1 + 1)}} |$
squaring we get,
$6 (x^{2} + y^{2} - 12 x - 14 y + 85) = x^{2} + y^{2} + 4 + 2 x y + 4 y + 4 x$
or $5 x^{2} + 5 y^{2} - 76 x - 88 y - 2 x y + 506 = 0.$

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