Question

The distance between the foci of the hyperbola $x^2 - 3y^2 - 4x - 6y -11 = 0$ is

• A. 4
• B. 6
• C. 8
• D. 10

Answer 1

Answer: (C)

$x^{2}-3 y^{2}-4 x-6 y-11=0$
$\Rightarrow \left(x^{2}-4 x\right)-3\left(y^{2}+2 y\right)=11$
$\Rightarrow \left(x^{2}-4 x+4\right)-3\left(y^{2}+2 y+1\right)=11+1$
$\Rightarrow \frac{(x-2)^{2}}{12}-\frac{(y+1)^{2}}{4}=1$
$\Rightarrow a^{2}=12, b^{2}=4$
$\therefore e=\sqrt{1+\frac{b^{2}}{a^{2}}}=\frac{2}{\sqrt{3}}$
Therefore distance between focii is
$=2 ae =2 \times 2 \sqrt{3} \times \frac{2}{\sqrt{3}}$
$=8$