A right angle prism $(45^{\circ} - 90^{\circ}- 45^{\circ})$ of refractive index n has a plate of refractive index $n_1\;( n_1 < n ) $ to its diagonal wall. The assembly is in air. A ray is incident at AB . Then the angle of incidence at AB for which the ray strikes the diagonal face at critical angle is :-

Question

$(a)\;\sin ^{-1} \frac{1}{\sqrt 2} \bigg[n^2-n_1^2)^{1/2}-n_1\bigg] \\ (b)\;\sin ^{-1} \frac{1}{\sqrt 2} \bigg[\sqrt {n^2-n_1^2}\bigg] \\ (c)\;\sin ^{-1} \bigg[\sqrt {n^2-n_1^2}-n_1\bigg] \\ (d)\;None$