The word ``elastically" implies that both energy and momentum are conserved. Assume, the incoming photon is incident in the direction and scatters in the y direction. Then conservation of momentum gives:
where and are the x and y components of the protons momentum after the collision, and and are the photons wavelengths before and after the collision, respectively. The initial energy is:
The final energy is
Equating the initial and final energy, we have that
Square both sides and solve for to find that
So, answer (D) is correct.