The Equation of the Ellipse with x + y + 2 = 0 as its direction one of its focus at (1, –1) and having eccentricity 2/3 is

$$\begin{gathered} \mathrm{let} \mathrm{p}(\mathrm{x}, \mathrm{y}) be \mathrm{Locus} \mathrm{of} \mathrm{point} on ellipse then \\ \mathrm{SP}=\mathrm{e}\left(\mathrm{pm}\right) \\ {\mathrm{SP}}^2={\mathrm{e}}^2 {\left(\mathrm{pm}\right)}^2 \\ \Rightarrow {(\mathrm{x}-1)}^2+{(\mathrm{y}+1)}^2 =\frac{4}{9}{\left|\frac{\mathrm{x}+\mathrm{y}+2}{\sqrt{1+1}}\right|}^2 \\ {\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+2\mathrm{y}+2 \\ =\frac{2}{9}({\mathrm{x}}^2+{\mathrm{y}}^2+4+2\mathrm{xy}+4\mathrm{x}+4\mathrm{y}) \\ 9{\mathrm{x}}^2+9{\mathrm{y}}^2-18\mathrm{x}+18\mathrm{y}+18 \\ =2{\mathrm{x}}^2+2{\mathrm{y}}^2+8+4\mathrm{xy}+8\mathrm{x}+8\mathrm{y} \\ 7{\mathrm{x}}^2+7{\mathrm{y}}^2-4\mathrm{xy}-26\mathrm{x}+10\mathrm{y}+10=0 \end{gathered}$$