Equation of the rectangular hyperbola whose focus is (1,-1) and the corresponding directrix is x-y+1=0 is

Question

Equation of the rectangular hyperbola whose focus is (1,-1) and the corresponding directrix is x-y+1=0 is (A) x2-y2=1 (B) x y=1 (C) 2 x y-4 x+4 y+1=0 (D) 2 x y+4 x-4 y-1=0

Tardigrade Admin · Accepted Answer

Eccentricity of rectangular hyperbola is √2. ∴ Equation of hyperbola is √(x-1)2+(y+1)2=√2 (|x-y+1|/√12+(-1)2) or (x-1)2+(y+1)2=(x-y+1)2 or 2 x y-4 x+4 y+1=0

Q. Equation of the rectangular hyperbola whose focus is $(1, - 1)$ and the corresponding directrix is $x - y + 1 = 0$ is

$x^{2} - y^{2} = 1$

$x y = 1$

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

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

Eccentricity of rectangular hyperbola is $2$ .
$∴$ Equation of hyperbola is
$(x - 1)^{2} + (y + 1)^{2} = 2 \frac{∣ x - y + 1∣}{1 ^{2} + ( - 1 ) ^{2}}$
or $(x - 1)^{2} + (y + 1)^{2} = (x - y + 1)^{2}$
or $2 x y - 4 x + 4 y + 1 = 0$

Q. Equation of the rectangular hyperbola whose focus is (1,−1) and the corresponding directrix is x−y+1=0 is

x2−y2=1

xy=1

2xy−4x+4y+1=0

2xy+4x−4y−1=0