Question

The equation of the ellipse with focus $(-1,1)$, directrix $x-y+3=0$ and eccentricity $\frac{1}{2}$ is

• A. $7x^2+2xy+7y^2+10x+10y+7=0$
• B. $7x^2+2xy+7y^2+10x-10y+7=0$
• C. $7x^2+2xy+7y^2+10x+10y-7=0$
• D. None of these

Answer 1

The correct option is B $7x^2+2xy+7y^2+10x-10y+7=0$

Given focus is $(1,-1)$ and equation of directrix is $x-y=3$

Eccentricity $e=\frac{1}{2}$

Hence $SP=ePM$

$S{P}^2=e^{2}P{M}^2$

i.e $S{P}^2=\frac{1}{4}(PM{)}^2$

$4S{P}^2=P{M}^2$

ie

$4\left[\right(x+1{)}^2+(y-1{)}^2]={\left(\frac{x-y-3}{\sqrt{1^2+(-1{)}^2}}\right)}^2$

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

on simplifying we get

$7x^2+2xy+7y^2+10x-10y+7=0$

Required equation of ellipse.