Question

The equation of the conic with focus at $(1, -1)$ directrix along $x - y + 1 = 0$ and with eccentricity $\sqrt{2}$ is

• A. $x^2 - y^2 = 1$
• B. $xy = 1$
• C. $2xy - 4x + 4y + 1 = 0$
• D. $2xy + 4x - 4y - 1 = 0$

Answer 1

Answer: (C)

Let $P (x, y)$ be any point on the conic. Then,
$\sqrt{\left(x -1\right)^{2} + \left( y + 1\right)^{2}} = \sqrt{2} \left(\frac{x - y + 1 }{\sqrt{2}}\right) $
$\Rightarrow \left(x - 1\right)^{2} + \left( y +1\right)^{2} = \left(x - y + 1\right)^{2}$
$ \Rightarrow 2xy - 4 x + 4y + 1 = 0 $