Q98 / 160 · AP EAMCET 2008 · Mathematics
The distance between the foci of the hyperbola $x^2-3 y^2-4 x-6 y-11=0$ is
- A$4$
- B$6$
- Correct answerC$8$
- D$10$
Solution
Given, equation of hyperbola is
$$
\begin{gathered}
x^2-3 y^2-4 x-6 y-11=0 \\
\Rightarrow \quad\left(x^2-4 x+4\right)-3\left(y^2+2 y+1\right)-11 \\
=4-3 \\
\Rightarrow \quad(x-2)^2-3(y+1)^2=12 \\
\Rightarrow \quad \frac{(x-2)^2}{12}-\frac{(y+1)^2}{4}=1 \\
\text { Now, } \quad e=\sqrt{1+\frac{4}{12}}=\frac{2}{\sqrt{3}}
\end{gathered}
$$
Now,
$$
e=\sqrt{1+\frac{4}{12}}=\frac{2}{\sqrt{3}}
$$
$\therefore$ Distance between foci
$$
=2 a e=2 \times \sqrt{12} \times \frac{2}{\sqrt{3}}=8
$$
Hence, the correct answer is option: 'C'.