Question

find the equation of the ellipse whose Centre is at the origin and foci are (1, 0) and (-1, 0) and eccentricity is 1/2​

Answer 1

let focus S(1,o) directrix x+ y +1 = 0 e = 1/root2. let P(x,y) be any point. M a point on the direc where the perpendicular from s meets the dir.

for an ellipse SP/PM = e

SP2 / PM2 = e2

( (x-1)2 + y2 ) / (x+y+1)2/2 = 1/2

x2 - 2x + 1 + y2 = 1/4 ( x2+y2+ 1 +2xy+2x+2y)C

4x2 - 8x + 4 + 4y2 = ( x2+y2+ 1 +2xy+2x+2y)

3x2 + 3y2 -2xy - 10x -2y + 3 = 0