Question

The equation to the ellipse, whose focus is the point (–1, 1), whose directrix is the straight line x – y + 3 = 0, and whose eccentricity is $\frac{1}{2}$ is –

• A. 7x² + 2xy + 7y² + 10x – 10y + 7 = 0
• B. x² + 2xy + 10x – 10y + 3 = 0
• C. 3x² + xy + 10x – 10y + 3 = 0
• D. None of these

Answer 1

Answer: (A)

7x² + 2xy + 7y² + 10x – 10y + 7 = 0

Answer 2

If P(x, y) be any point on the ellipse, S be its focus, and PN be the perpendicular from P on directrix, then by definition of an ellipse PS² = e² PN², hence

(x + 1)² + (y – 1)² = $\frac { 1 } { 4 }$ $\left( \frac{x - y + 3}{\sqrt{2}} \right)^{2}$=$\frac{(x - y + 3)^{2}}{8}$

[As focus is (–1, 1) and directrix is x – y + 3 = 0]

Ž 8(x² + y² + 2x – 2y + 2)

= x² + y² + 9 – 2xy + 6x – 6y7x² + 2xy + 7y² + 10x – 10y + 7 = 0.

Hence (1) is the correct answer.