The asymptote of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ form with any tangent

Question

The asymptote of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ form with any tangent to the hyperbola a triangle whose area is $a^{2}tan\lambda$ in magnitude then its eccentricity is:

Accepted Answer

Sol. (a)equation of tangent is$$\frac{xsec	heta}{a} - \frac{ytan	heta}{b} = 1$$Solving point of intersection of tangent with asymptotes we get,$$A \equiv a(sec	heta + tan	heta),b(sec	heta + tan	heta) $$$B \equiv a(sec	heta - tan	heta), - b(sec	heta - tan	heta)$ Area of$$\begin{matrix} & \ \ & \ = ab \ & \ = ab = a^{2}tan\lambda \ \Rightarrow & \frac{b^{2}}{a^{2}} = 	an^{2}\lambda \ \Rightarrow & e^{2} - 1 = 	an^{2}\lambda \ \Rightarrow & e = sec\lambda \end{matrix}$$

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