Question

The equation of the hyperbola having its eccentricity $2$ and the distance between its foci is $8$, is:

• A. $\frac{x^2}{12}-\frac{y^2}{4}=1$
• B. $\frac{x^2}{4}-\frac{y^2}{12}=1$
• C. $\frac{x^2}{8}-\frac{y^2}{2}=1$
• D. $\frac{x^2}{16}-\frac{y^2}{9}=1$

Answer 1

The correct option is B

$\frac{x^2}{4}-\frac{y^2}{12}=1$

Explanation for the correct answer :

Given that eccentricity is $2$ and distance between its foci is $8$.

$\therefore e=2,2ae=8$

Calculating $a$:

$$\begin{gathered} 2ae=8 \\ \Rightarrow 2a\times 2=8\left[\text{Substituting}e=2\right] \\ \Rightarrow a=2 \end{gathered}$$

Calculating $b$:

$$\begin{gathered} b^2=a^2(e^2-1) \\ \Rightarrow b^2=2^2(2^2-1) \\ \Rightarrow b^2=4\times 3 \\ \Rightarrow b^2=12 \end{gathered}$$

Hence, the equation of the hyperbola is

$$\begin{gathered} \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\ \Rightarrow \frac{x^2}{4}-\frac{y^2}{12}=1 \end{gathered}$$

Therefore, the correct answer is option (B).