A normal to the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2 = 1 meets the axis...

Normal to the hyperbola is

$axcos \theta + bycot \theta = a^{2} + b^{2}$

Putting $y = 0$ and $x = 0$

We get A \left(\dfrac{a^{2} + b^{2}}{a}sec $\theta,$0\right), B\left($0,$\dfrac{a^{2} + b^{2}}{b}$tan$\theta\right)

line through $A$ , perpendicular to the $x$ -axis is $x =$\dfrac{a^{2} + b^{2}}{a}$sec$\theta

line through $B$ , perpendicular to the $y$ -axis is $y =$\dfrac{a^{2} + b^{2}}{b}$tan$\theta

So intersection of these point is $L$

$∵ \sec^{2}$\theta - \tan^{2} \theta =$1$

$⇒ a^{2}x^{2} - b^{2}y^{2} = {\left(a^{2} + b^{2}\right)}^{2}$ .