The equation of the ellipse with $x + y + 2 = 0$ as its directrix, one of its focus at $(1, - 1)$ and having eccentricity 2/3 is.

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$7 x^{2} + 7 y^{2} + 4 x y + 26 x + 26 y + 10 = 0$

$7 x^{2} + 7 y^{2} - 4 x y - 26 x + 10 y - 10 = 0$

$7 x^{2} + 7 y^{2} - 4 x y + 26 x + 26 y - 10 = 0$

$7 x^{2} + 7 y^{2} + 4 x y - 26 x - 26 y + 10 = 0$

answer is B.

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Detailed Solution

Given $S = (1, - 1), e = \frac{2}{3}, equation of directrix is x + y + 2 = 0$

Let $P = (x_{1}, y_{1})$ be a point on the ellipse

$\begin{array}{l} ∴ \frac{S P}{P M} = e \Rightarrow S P = e P M \Rightarrow \sqrt{{(x_{1} - 1)}^{2} + {(y_{1} + 1)}^{2}} = \frac{2}{3} \frac{|x_{1} + y_{1} + 2|}{\sqrt{2}} \\ \Rightarrow 9 [x_{1}^{2} + y_{1}^{2} - 2 x_{1} + 2 y_{1} + 2] = 2 (x_{1}^{2} + y_{1}^{2} + 4 + 2 x_{1} y_{1} + 4 y_{1} + 4 x_{1}) \\ \Rightarrow 7 x_{1}^{2} - 4 x_{1} y_{1} + 7 y_{1}^{2} - 26 x_{1} + 10 y_{1} - 10 = 0 \end{array}$