Question

Find the equation of the ellipse whose focus is $(1,0)$, the directrix is $x+y+1=0$ and eccentricity is $\frac{1}{\sqrt{2}}$.

Answer 1

Let $S$ be the focus of the ellipse and $e$ be the eccentricity of the ellipse.

Consider that $P\left(x,y\right)$ be any point on the ellipse, then by the definition of ellipse,

$SP=e\times PM$

$\sqrt{{\left(x-1\right)}^2+{\left(y-0\right)}^2}=\frac{1}{\sqrt{2}}\left(\frac{x+y+1}{\sqrt{1^2+1^2}}\right)$

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

${\left(x-1\right)}^2+{\left(y\right)}^2=\frac{1}{4}{\left(x+y+1\right)}^2$

$3x^2+3y^2-2xy-10x-2y+3=0$

The required equation of ellipse is $3x^2+3y^2-2xy-10x-2y+3=0$.