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 $x^{2}+y^{2}=a^{2}-b^{2}$
$\Rightarrow 7=a^{2}-b^{2}\Rightarrow a^{2}=16,b^{2}=9$
$\Rightarrow \left|\frac{b}{a}\right|=\frac{3}{4}=0.75$