$\text {Momentum of particle 1 }p_1 = \frac{h}{ \lambda_1}$

$ \text{velocity of particle 2 } p_2 = \frac{h}{\lambda_2}$

$ \text {Total Initial momentum = Total final momentum} P = \sqrt{p_1^2 + p_2^2} $

$\text{Final wavelength }\lambda = \frac{h}{P} = \frac{h}{\sqrt{p_1^2 + p_2^2}} $

$ \frac{1}{\lambda^2} = \frac{1}{\lambda_1^2} + \frac{1}{\lambda_2^2}$