Focus (–1, 1),  $\mathrm{e}=\frac{1}{2}$ directirx is x – y + 3 = 0 eq. of the ellipse is 


 

$$\begin{gathered} \mathrm{S}=(-1, 1) \mathrm{e}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ \mathrm{L}=\mathrm{x}-\mathrm{y}+3=0 \\ Let \mathrm{P}(\mathrm{x}, \mathrm{y}) be a point on hyperbola then \\ \frac{\mathrm{SP}}{\mathrm{PM}}=\mathrm{e} \Rightarrow {\mathrm{SP}}^2={\mathrm{e}}^2{\left(\mathrm{PM}\right)}^2 \\ {(\mathrm{x}+1)}^2+{(\mathrm{y}-1)}^2=\frac{1}{4}{\left|\frac{\mathrm{x}-\mathrm{y}+3}{\sqrt{1+1}}\right|}^2 \\ 8({\mathrm{x}}^2+1+2\mathrm{x}+{\mathrm{y}}^2+1-2\mathrm{y}) \\ ={\mathrm{x}}^2+{\mathrm{y}}^2+9-2\mathrm{xy}-6\mathrm{y}+6\mathrm{x} \\ 7{\mathrm{x}}^2+2\mathrm{xy}+7{\mathrm{y}}^2+10\mathrm{x}-10\mathrm{y}+7=0 \end{gathered}$$