Two particles move at right angle to each other. Their de Broglie wavelengths are $$\lambda_1$$ and $$\lambda_2$$ respectively. The particles suffer perfectly inelastic collision. The de Broglie wavelength $$\lambda$$ of the final particle, is given by:

First recall the de Broglie relation, which connects the momentum $$p$$ of a particle with its wavelength $$\lambda$$:

$$\lambda = \dfrac{h}{p},$$

where $$h$$ is Planck’s constant.

Let the two particles have momenta $$p_1$$ and $$p_2$$, and let their de Broglie wavelengths be $$\lambda_1$$ and $$\lambda_2$$ respectively. Using the above formula we write

$$p_1 = \dfrac{h}{\lambda_1}, \qquad p_2 = \dfrac{h}{\lambda_2}.$$

The question tells us that the particles move at right angles to each other, so the angle between the two momentum vectors is $$90^{\circ}$$. Because the collision is perfectly inelastic, the particles stick together and move as a single combined particle. Linear momentum is conserved, and since the original momenta are perpendicular, the magnitude of the final momentum $$P$$ is obtained from the Pythagorean theorem:

$$P = \sqrt{p_1^{\,2} + p_2^{\,2}}.$$

Substituting $$p_1 = \dfrac{h}{\lambda_1}$$ and $$p_2 = \dfrac{h}{\lambda_2}$$ into this expression gives

$$P = \sqrt{\left(\dfrac{h}{\lambda_1}\right)^{\!2} + \left(\dfrac{h}{\lambda_2}\right)^{\!2}} = h \,\sqrt{\dfrac{1}{\lambda_1^{\,2}} + \dfrac{1}{\lambda_2^{\,2}}}.$$

If $$\lambda$$ is the de Broglie wavelength of the composite particle after the collision, we again use the de Broglie relation, this time for the final particle:

$$\lambda = \dfrac{h}{P}.$$

Substituting the value of $$P$$ that we have just obtained, we get

$$\lambda \;=\; \dfrac{h}{\,h \,\sqrt{\dfrac{1}{\lambda_1^{\,2}} + \dfrac{1}{\lambda_2^{\,2}}}} \;=\; \dfrac{1}{\sqrt{\dfrac{1}{\lambda_1^{\,2}} + \dfrac{1}{\lambda_2^{\,2}}}}.$$

Taking the reciprocal of both sides and then squaring, we arrive at

$$\dfrac{1}{\lambda^{\,2}} \;=\; \dfrac{1}{\lambda_1^{\,2}} \;+\; \dfrac{1}{\lambda_2^{\,2}}.$$

This is exactly the relation stated in Option D.

Hence, the correct answer is Option D.