Question

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

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

Answer 1

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

Given: focus = S$(-1,1),$ directrix is $x-y+3=0$

and eccentricity $=\frac{1}{2}$

Let any point on the ellipse be $P(x,y)$

From definition of ellipse, we write

$SP=e\cdot PM$

Where $PM$ is perpendicular distance from P to the directrix

$\Rightarrow \sqrt{(x+1{)}^2+(y-1{)}^2}=\frac{1}{2}\times \left|\frac{x-y+3}{\sqrt{1^{2+(-1{)}^2}}}\right|$

Squaring on both sides, we get

$\Rightarrow (x+1{)}^2+(y-1{)}^2=\frac{1}{4}\times \frac{(x-y+3{)}^2}{2}$

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

$=\frac{1}{8}(x^2+y^2+9+6x-6y-2xy)$

$\Rightarrow 8x^2+16x+8y^2-16y+16$

$=x^2+y^2+9+6x-6y-2xy$

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

So, the equation of ellipse is

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

Hence, option (B) is correct.