A tangent drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at  $P(a\sec \frac{\pi }{6},b\tan \frac{\pi }{6})$ form a triangle of area  $3a^2$  sq.units with the coordinate axes. The eccentricity of the conjugate hyperbola of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$  is

 $P=(\frac{2a}{\sqrt{3}},\frac{b}{\sqrt{3}})$
Equation of the tangent at P is  $\frac{2x}{\sqrt{3}a}-\frac{y}{\sqrt{3}b}=1$
X intercept & y intercept of the tangent are $\frac{\sqrt{3}a}{2}\&-\sqrt{3}b$   respectively
Area of the triangle formed by the tangent with the coordinate axes is
 $\frac{1}{2}\times \frac{\sqrt{3}a}{2}\times \sqrt{3}b=3a^2\Rightarrow \frac{a}{b}=\frac{1}{4}$
The eccentricity of the conjugate hyperbola equals to
 $\sqrt{1+\frac{1}{16}}=\frac{\sqrt{17}}{4}$