Question

For the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ distance between the foci is $10$ units. From the point $\left(2,\sqrt{3}\right),$ perpendicular tangents are drawn to the hyperbola, then the value of $\left|\frac{b}{a}\right|$ is

• A. $0.25$
• B. $5$
• C. $0.75$
• D. $1$

Answer 1

Answer: (C)

Let, $e$ be the eccentricity of the hyperbola
Now, $2ae=10\Rightarrow a^2{e}^2=25$
$\Rightarrow a^2+b^2=25$
Also $\left(2,\sqrt{3}\right)$ lies on the director circle $x^2+y^2=a^2-b^2$
$\Rightarrow 7=a^2-b^2\Rightarrow a^2=16,b^2=9$
$\Rightarrow \left|\frac{b}{a}\right|=\frac{3}{4}=0.75$