Question

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

• A. $7x^2+2xy+7y^2+7=0$
• B. $7x^2+2xy+7y^2–10x+10y+7=0$
• C. $7(x^2+y^2)+2xy+10x–10y–7=0$
• D. $7(x^2+y^2)+2xy–10x–10y+7=0$

Answer 1

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

Focus: $S=(1,-1)$
Directrix equation: $x-y-3=0$
Eccentricity $e=\frac{1}{2}$
Let $P(x,y)$ be any point on the ellipse then from the defination of ellipse,
$\frac{SP}{PM}=e$
Where $PM$ is the distance from the point $P$ on the directrix,

$\sqrt{(x-1{)}^2+(y+1{)}^2}=\frac{1}{2}\left|\frac{x-y-3}{\sqrt{1^2+(-1{)}^2}}\right|$
$\Rightarrow (x-1{)}^2+(y+1{)}^2=\frac{1}{8}(x-y-3{)}^2\Rightarrow 8(x^2+y^2-2x+2y+2)=x^2+y^2-2xy-6x+6y+9$
$\Rightarrow 7x^2+2xy+7y^2–10x+10y+7=0$