Question

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

• A. $7x^2+2xy+7y^2-10x+10y+7=0$
• B. $7x^2+2xy+7y^2+7=0$
• C. $7x^2+2xy+7y^2+10x-10y-7=0$
• D. $Noneofthese$

Answer 1

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

equation of ellipse

focus $(+1,-1)$

line $\to x-1=3$

$e=1/2$

$\to$ Given focus (1, - 1) & equation x - y = 3

SP = e.PM

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

$\therefore S{P}^2=\frac{1}{4}P{M}^2$

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

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

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

$\therefore 8(x^2+y^2+2x-2y+2)=x^2+y^2+9+2\left(x\right)(-y)+2(-y)\left(3\right)+2\left(3x\right)$

$\therefore 7x^2+2x+7y^2+10x-10y+7=0$