Question

The equation of the ellipse whose focus is (1, –1), the directrix the line x – y – 3 = 0 and eccentricity 1/2 is

A. 7x² + 2xy + 7y² – 10x + 10y + 7 = 0
B. 7x² + 2xy + 7y² + 7 = 0
C. 7x² + 2xy + 7y² + 10x – 10y – 7 = 0
D. none

Answer 1

Focus (S) = (1, -1) Directrix is x – y – 3 = 0 & e = 1/2

As, for any point P (x,y) on the ellipse,

Distance from focus to the point (x,y) = e (Distance from (x,y) to the foot of perpendicular on directrix)

Perpendicular Distance (Between a point and line) = , whereas the point is (x₁, y₁) and the line is expressed as ax + by + c = 0

Squaring both the sides,

8[x² - 2x + 1 + y² + 2y + 1] = x² - 6x - 2xy + y² + 6y + 9

8x² - 16x + 16 + 8y² + 16y = x² - 6x - 2xy + y² + 6y + 9

7x² - 10x + 2xy + 7y² + 10y + 7 = 0

Hence the required equation is 7x² - 10x + 2xy + 7y² + 10y + 7 = 0

OPTION (A) is the correct answer.