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)

$2xy - 4x + 4y + 1 = 0$

Answer 2

Here, focus (S) = (1, –1), eccentricity (5)=$\sqrt{2}$

From definition , $SP = ePM$

$\sqrt{(x - 1)^{2} + (y + 1)^{2}} = \frac{\sqrt{2}.(x - y + 1)}{\sqrt{1^{2} + 1^{2}}}$⇒ $(x - 1)^{2} + (y + 1)^{2}$ =

$(x - y + 1)^{2}$ ⇒ $2xy - 4x + 4y + 1 = 0$, which is the required equation of conic (Rectangular hyperbola)