Question

The equation of an ellipse whose focus is (–1, 1), whose directrix is $x - y + 3 = 0$and whose eccentricity is $\frac{1}{2},$ is given by

• A. $7x^{2} + 2xy + 7y^{2} + 10x - 10y + 7 = 0$
• B. $7x^{2} - 2xy + 7y^{2} - 10x + 10y + 7 = 0$
• C. $7x^{2} - 2xy + 7y^{2} - 10x - 10y - 7 = 0$
• D. $7x^{2} - 2xy + 7y^{2} + 10x + 10y - 7 = 0$

Answer 1

Answer: (A)

$7x^{2} + 2xy + 7y^{2} + 10x - 10y + 7 = 0$

Answer 2

Let any point on it be $(x,y)$ then by definition,

$\sqrt{\mathbf{(x + 1}\mathbf{)}^{\mathbf{2}}\mathbf{+ (y}\mathbf{-}\mathbf{1}\mathbf{)}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left| \frac{\mathbf{x}\mathbf{-}\mathbf{y + 3}}{\sqrt{\mathbf{1}^{\mathbf{2}}\mathbf{+}\mathbf{1}^{\mathbf{2}}}} \right|$

Squaring and simplifying, we get

$7x^{2} + 2xy + 7y^{2} + 10x - 10y + 7 = 0$, which is the required ellipse