Let  $P(x_1,y_1)$  be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$  , the tangent at point P, passes through $Q(0,-b)$ & the normal at point P, passes through the point $R(2\sqrt{2}a,0)$ . Let e be the eccentricity of the hyperbola then the value of  $e^2=\_\_\_\_\_$

 tangent at $P(x_1,y_1)$  in $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$  Since it passes through $(0,-b)$ $\Rightarrow y_1=b$    
 Equation of normal at  $P(x_1,y_1)$  is  $\frac{a^{2}x}{x_1}+\frac{b^{2}y}{y_1}=a^2{e}^2$
 Since it passes through $(2\sqrt{2}a,0) \Rightarrow x_1=\frac{2\sqrt{2}a}{e^2}$  
$(\frac{2\sqrt{2}a}{e^2},b)$   lies on hyperbola  $\Rightarrow \frac{8a^2}{e^4{a}^2}-\frac{b^2}{b^2}=1 \Rightarrow e^2=2$