A charged particle $X$ is moving towards another charged particle $Y$. For the ‘$X$ plus $Y$’ system, the total momentum is $p$ and the total energy is $E$. (This question has multiple correct options)
(A). $p$ and $E$ are conserved if both $X$ and $Y$ are free to move
(B). (A) is true only if $X$ and $Y$ have similar charges
(C). If $Y$ is fixed, $E$ is conserved but not $p$
(D). If $Y$ is fixed, neither $E$ is conserved nor $p$

Hint: Two charged particles make a system. Since, $X$ is moving towards the particle $Y$, there must be a force of attraction between them. The system of charged particles is isolated as long as no external force acts on the particles. Momentum is conserved when the system is isolated and energy is conserved when no force does any work on the system.

Complete step by step solution:
Both $X$ and $Y$ are charged particles, so an electrostatic force of attraction or repulsion will exist between them but since it acts between the charges, it is an internal force. Therefore, the ‘$X$ plus $Y$’ system is an isolated system if they are free to move and no external force is acting on them.
The charges on the particles do not matter as the electrostatic forces are internal forces. So, no matter what the charge, the system will remain isolated unless an external force acts on them.
When we fix the position of $Y$, an external force is now acting on the system which binds $Y$ to its position. Therefore, the system is no longer isolated and the momentum is not conserved. But as the external force does not work on the particle $Y$ or the system, the energy of the system is conserved.

As $p$ and $E$ are conserved as long as the system is isolated, when $Y$ is fixed, $p$ is not conserved but $E$ is conserved, the correct options are (A) and (C).

Note:
The electrostatic force is the force of attraction or repulsion between two charged particles. It acts along the line joining both the particles. The law of conservation of momentum states that the sum of initial momentum of bodies is equal to the sum of final momentum. The mechanical energy of a system remains constant when energy is conserved.