Question

The equation of the hyperbola whose vertices are $(\pm 6,0)$ and one of its directrix is $x=4$ is

• A. $\frac{x^2}{45}-\frac{y^2}{36}=1$
• B. $\frac{x^2}{36}-\frac{y^2}{45}=1$
• C. $\frac{x^2}{25}-\frac{y^2}{29}=1$
• D. $\frac{y^2}{25}-\frac{x^2}{36}=1$

Answer 1

The correct option is B $\frac{x^2}{36}-\frac{y^2}{45}=1$

Directrix are given by $x=\pm \frac{a}{e}\cdots \left(1\right)$
Distance between the center and vertices $=a=6$,
Now, putting value of $a$ in $\left(1\right)$
$4=\frac{6}{e}$
$\Rightarrow e=\frac{3}{2}$
$\Rightarrow \frac{x^2}{a^2}-\frac{y^2}{a^2(e^2-1)}=1$
$\Rightarrow \frac{x^2}{36}-\frac{y^2}{45}=1$