Find the standard equation of ellipse whose focus is (1,0), the directrix is x+y+1=0 and eccentricity is 1/root2

For an ellipse,Given, focus S=1,0 and equation of directrix isx+y+1=0 and eccentricity e=12Let P be any point x,y. So, by definition of an ellipse,SP=e×PMx-12+y-02=12×x+y+112+12x-12+y2=12x+y+1Squaring on both sides, we have,x-12+y2=14x+y+124x-12+y2=x+y+124x2+1-2x+y2=x2+y2+1+2xy+2y+2x4x2+4-8x+4y2=x2+y2+1+2xy+2y+2x3x2+3y2-2xy-10x-2y+3=0

  • 67

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

  • 47
What are you looking for?