Question

The angle between asymptotes of the standard hyperbola (centered at origin and $x$-axis as transverse axis) is $\frac{\pi }{3}$. If the radius of the director circle of the hyperbola is $4$ then the equation of the hyperbola is

• A. $\frac{x^2}{3}-\frac{y^2}{1}=8$
• B. $\frac{x^2}{25}-\frac{y^2}{9}=1$
• C. $\frac{x^2}{2}-\frac{y^2}{1}=8$
• D. $\frac{x^2}{25}-\frac{y^2}{16}=1$

Answer 1

The correct option is A $\frac{x^2}{3}-\frac{y^2}{1}=8$

Let the equation of the hyperbola is
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
then the equation of the director circle of the hyperbola is
$x^2+y^2=a^2-b^2\Rightarrow a^2-b^2=16$
$2{\tan }^{-1}\frac{b}{a}=\frac{\pi }{3}\frac{b}{a}=\frac{1}{\sqrt{3}}{b}^2=8,a^2=24$