Question

The equation of the conjugate hyperbola of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$ is

• A. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+2=0$
• B. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+5=0$
• C. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+3=0$
• D. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-2=0$

Answer 1

The correct option is B $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+5=0$

Conjugate hyperbola for $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$

Equation can be written as $(x^2-2\sqrt{5}x+5)-2(y^2+2\sqrt{2}y+2)=3+5+(-4)$
$\Rightarrow \frac{(x-\sqrt{5}{)}^2}{4}-\frac{(y+\sqrt{2}{)}^2}{2}=1$

Conjugate hyperbola is $\frac{(x-\sqrt{5}{)}^2}{4}-\frac{(y+\sqrt{2}{)}^2}{2}=-1$
$\Rightarrow x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+5=0$