Question

Find the equation of the ellipse whose eccentricities is $\frac{1}{2}$ , focus is $(-1,1)$ . directrix is $y=x+3.$

Answer 1

$\frac{PS}{PM}=e$

$\therefore \frac{\sqrt{(x+1{)}^2+(y-1{)}^2}}{\left|\frac{x-y+3}{\sqrt{1+1}}\right|}=\frac{1}{2}$

$\Rightarrow (x+1{)}^2+(y-1{)}^2=\frac{1}{4}\frac{(x-y+3{)}^2}{(\sqrt{2}{)}^2}$

$\Rightarrow 8\left(\right(x+1{)}^2+(y-1{)}^2)=(x-y+3{)}^2$.