For a particle projected horizontally, at any point horizontally component of the velocity is eqau to the horizontal velocity of projection.

$\mathrm{u}={\mathrm{V}}_1\cos {37}^{\mathrm{o}}={\mathrm{V}}_2\cos {53}^{\mathrm{o}}\text{ and }$

de Broglie wavelength of a particle is 

$\mathrm{\lambda }=\frac{\mathrm{h}}{\mathrm{mV}}$

$\Rightarrow \frac{{\mathrm{\lambda }}_1}{{\mathrm{\lambda }}_2}=\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1}=\frac{\cos {37}^0}{\cos {53}^0}=\frac{4}{3}$