find the equation of the ellipse whose focus is (-1 , 1), directrix is x + y + 3 = 0 and eccentricity e = 1/2.

Question

find the equation of the ellipse whose focus is (-1 , 1),
directrix is x + y + 3 = 0 and eccentricity e = 1/2

Priyanka Kedia · Accepted Answer

Given, focus S-1,1, Directrix=x+y+3=0 and eccentricity e=12Let Px,y
be any point on ellipse and PM is perpendicular to the directrix,
thenSP=ePM⇒SP2=e2 PM2⇒x+12+y-12=14x+y+312+122
⇒8x2+1+2x+y2+1-2y=x+y+32
⇒8x2+2+2x+y2-2y=x2+y2+2xy+9+6x+6y⇒8x2+16+16x+8y2-16y=x2+y2+2xy+9+6x+6y⇒7x2+7y2-2xy+10x-24y+7=0