Question

If the length of the latusrectum and the length of transverse axies of a hyperbola are $4\sqrt{3}$ and $2\sqrt{3}$ respectively, then the equation of the hyperbola is

• A. $\frac{x^2}{3}-\frac{y^2}{4}=1$
• B. $\frac{x^2}{3}-\frac{y^2}{9}=1$
• C. $\frac{x^2}{6}-\frac{y^2}{9}=1$
• D. $\frac{x^2}{6}-\frac{y^2}{3}=1$
• E. $\frac{x^2}{3}-\frac{y^2}{6}=1$

Answer 1

The correct option is E $\frac{x^2}{3}-\frac{y^2}{6}=1$

Length if transverse axis $=2a$

$2a=2\sqrt{3}\Rightarrow a=\sqrt{3}$

Length of Latusrectum $=\frac{2b^2}{a}$

$\frac{2b^2}{a}=4\sqrt{3}\Rightarrow b^2=2\sqrt{3}a\Rightarrow b^2=2\sqrt{3}\left(\sqrt{3}\right)=6$

Equation of hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

Substituting $a$ and $b$, we get

$\Rightarrow \frac{x^2}{3}-\frac{y^2}{6}=1$

So, option E is correct.