If for a conic section a focus is $(- 1, 1)$ , eccentricity $= 3$ and the equation of the corresponding directrix is $x - y + 3 = 0$ then the equation of the conic section is

7 x^{2} - 18 x y + 7 y^{2} + 50 x - 50 y + 77 = 0

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7 x^{2} + 18 x y + 7 y^{2} = 1

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7 x^{2} + 18 x y + 7 y^{2} - 50 x + 50 y + 77 = 0

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none of these

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Solution

The correct option is C $7 x^{2} - 18 x y + 7 y^{2} + 50 x - 50 y + 77 = 0$
Using definition of conic, $P S^{2} = e^{2} . P M^{2}$
$(x + 1)^{2} + (y - 1)^{2} = 3^{2} {∣ ∣ ∣ \frac{x - y + 3}{\sqrt{2}} ∣ ∣ ∣}^{2}$
$2 (x^{2} + y^{2} + 2 x - 2 y + 2) = 9 (x^{2} + y^{2} - 2 x y - 6 y + 6 x + 9)$
$7 x^{2} - 18 x y + 7 y^{2} + 50 x - 50 y + 77 = 0$

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