Find the equation of the ellipse with focus at (1, –1), e = 2/3 and directrix is  x + y + 2 = 0.

Given focus(S)= (1, –1), e = 2/3 and directrix is x + y + 2 = 0
Let P(x, y) be any point on ellipse
$\begin{array}{l}\Rightarrow \frac{\mathrm{SP}}{\mathrm{PM}}=\mathrm{e}\Rightarrow \mathrm{SP}=\mathrm{ePM}\\ \sqrt{(\mathrm{x}-1{)}^2+(\mathrm{y}+1{)}^2}=\frac{2}{3}\frac{|\mathrm{x}+\mathrm{y}+2|}{\sqrt{1+1}}\end{array}$
S.O.B.S
$\begin{array}{l}(\mathrm{x}-1{)}^2+(\mathrm{y}+1{)}^2=\frac{4}{9}\left[\frac{(\mathrm{x}+\mathrm{y}+2{)}^2}{2}\right]\\ 9\left({\mathrm{x}}^2+1-2\mathrm{x}+{\mathrm{y}}^2+1+2\mathrm{y}\right)=2\left({\mathrm{x}}^2+{\mathrm{y}}^2+4+2\mathrm{xy}+4\mathrm{y}+4\mathrm{x}\right)\\ 9{\mathrm{x}}^2+9-18\mathrm{x}+9{\mathrm{y}}^2+9+18\mathrm{y}-2{\mathrm{x}}^2-2{\mathrm{y}}^2-8-4\mathrm{xy}-8\mathrm{y}-8\mathrm{x}=0\\ 7{\mathrm{x}}^2-4\mathrm{xy}+7{\mathrm{y}}^2-26\mathrm{x}+10\mathrm{y}+10=0\end{array}$