Question

From a point $P(1,2)$ two tangents are drawn to a hyperbola $H$ in which one tangent is drawn to each arm of the hyperbola. If the equations of asymptotes of hyperbola $H$ are $\sqrt{3}x-y+5=0$ and $\sqrt{3}x+y-1=0$, then eccentricity of $H$ is :

• A. $2$
• B. $\frac{2}{\sqrt{3}}$
• C. $\sqrt{2}$
• D. $\sqrt{3}$

Answer 1

The correct option is B $\frac{2}{\sqrt{3}}$

Since $c_1{c}_2(a_1{a}_2+b_1{b}_2)<0$

$\therefore$ origin lies in acute angle.

$P(1,2)$ lies in obtuse angle

Slope of asymptotes$m_1=\sqrt{3},m_2=-\sqrt{3}$

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$=\left|\frac{\sqrt{3}-(-\sqrt{3})}{1+\sqrt{3}\times -\sqrt{3}}\right|$

$=\left|\frac{2\sqrt{3}}{1-3}\right|$

$=\left|\frac{2\sqrt{3}}{-2}\right|$

$\Rightarrow \tan \theta =\sqrt{3}$

Acute angle between the asymptotes is $\frac{\pi }{3}$

Hence eccentricity $e=\sec \frac{\theta }{2}=\sec \frac{\pi }{6}=\frac{2}{\sqrt{3}}$