Question

The equation of conic with focus $(1,-1)$ directrix along $x-y+1=0$ and eccentricity $\sqrt{2}$ is

Answer 1

$e=\sqrt{2},S(1,-1)$ and equation of directrix is $x-y+1=0$

Let a point $P(x,y)$, such that

$SP=ePM$

Where $PM$ is perpendicular distance from $P(x,y)$ to directrix

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

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

Squaring on both sides

$\Rightarrow (x-1{)}^2+(y+1{)}^2=(x-y+1{)}^2$

$\Rightarrow x^2-2x+1+y^2+2y+1$

$=x^2+y^2+1-2xy-2y+2x$

$\Rightarrow 2xy-4x+4y+1=0$

Equation of conic is $2xy-4x+4y+1=0$

Hence, option (D) is correct.