To find the value of charge q such that when it is placed at the center of two equal and like charges Q, the three charges are in equilibrium, we need to consider the electrostatic forces acting on each charge.

Let's break down the problem step by step:

1. Identify the forces:
- The charge q experiences a repulsive force from each of the charges Q.
- The two charges Q also experience a repulsive force from each other.

2. Equilibrium condition:
For the three charges to be in equilibrium, the net force on each charge must be zero. This means that the forces acting on each charge must cancel out.

3. Forces on charge q:
- The force on charge q due to one of the charges Q is given by Coulomb's law: F = k(qQ/r^2), where k is the electrostatic constant, r is the distance between the charges, and F is the force.
- Since there are two charges Q on either side of q, the total force on q is 2F.

4. Forces on charges Q:
- The force on one of the charges Q due to the other charge Q is also given by Coulomb's law: F = k(QQ/r^2).
- Since there is one charge Q on either side of q, the total force on each charge Q is F.

5. Equating the forces:
- The net force on q is 2F, and it must be equal to the force on each charge Q, which is F.
- Therefore, 2F = F.
- Cancelling out F from both sides, we get 2 = 1.

6. Conclusion:
- Since we have arrived at the contradictory statement 2 = 1, it implies that there is no value of charge q that can satisfy the equilibrium condition when it is placed at the center of two equal and like charges Q.

Therefore, none of the given options (a) Q/4, (b) Q, (c) -Q/4, or (d) Q/2 is the correct answer.