Question

2.The equation of ellipse whose focus is S(1,-1), directrix the line x – y -3 = 0 ans eccentricity ½ is

Answer 1

Answer:

The equation of the required ellipse is

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

Step-by-step explanation:

Given:

Focus S(1,-1) and directrix x-y-3=0

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

By definition of conic

$\frac{SP}{PM}=e$

$\frac{SP}{PM}=\frac{1}{2}$

$\implies\,2\,SP=PM$

$\implies\,2\sqrt{(x-1)^2+(y+1)^2}=\frac{x-y-3}{\sqrt{1+1}}$

$\implies\,2\sqrt{(x-1)^2+(y+1)^2}=\frac{x-y-3}{\sqrt2}$

squarring on both sides

$\implies\,4[(x-1)^2+(y+1)^2]=\frac{(x-y-3)^2}{2}$

$\implies\,8[(x-1)^2+(y+1)^2]=(x-y-3)^2$

$\implies\,8[x^2+1-2x+y^2+1+2y]=x^2+y^2+9-2xy+6y-6x$

$\implies\,8x^2-16x+8y^2+16+16y=x^2+y^2+9-2xy+6y-6x$

$\implies\boxed{\bf\,7x^2+2xy+7y^2-10x+10y+7=0}$