Let - 2y = μ be a tangent to the hyperbola a ^ 2 x ^ 2 - y ^ 2 = b...

Given that $\lambdax - 2y = \mu$ is a tangent to the curve ${a}^{2}{x}^{2} - {y}^{2} = {b}^{2}$

So ${a}^{2}{x}^{2} - {\left(\dfrac{\lambdax - \mu}{2}\right)}^{2} = {b}^{2}$

$⇒ \left(4{a}^{2} - {\lambda}^{2}\right){x}^{2} + 2\lambda\mux - {\mu}^{2} - 4{b}^{2} = 0$

So for the line to be tangent, the condition is discriminant $= 0$

i.e. $4{\lambda}^{2}{\mu}^{2} + 4\left(4{a}^{2} - {\lambda}^{2}\right)\left({\mu}^{2} + 4{b}^{2}\right) = 0$

$4{\lambda}^{2}{b}^{2} - 4{a}^{2}{\mu}^{2} = 16{a}^{2}{b}^{2}$

Hence, $\dfrac{{\lambda}^{2}}{{a}^{2}} - \dfrac{{\mu}^{2}}{{b}^{2}} = 4$