Solution:

The given problem can be solved by using the principle of superposition of electric fields or forces.

Step 1:
Let us first consider the electric field and the force on the charge q due to the two charges Q. Since the two charges Q are equal in magnitude and are placed at equal distances from q, the electric field due to each charge will be equal in magnitude and will be directed towards the charge q. Thus, the net electric field at q due to the two charges Q will be zero.

Similarly, the two forces acting on q due to the two charges Q will be equal in magnitude and will be directed towards q. Thus, the net force on q due to the two charges Q will also be zero.

Step 2:
Now, let us consider the electric field and the force on q due to itself. Since q is a point charge, it will produce an electric field that is spherically symmetric around it. Therefore, the electric field at the centre of the line joining the two charges Q due to q will be zero.

Similarly, the force on q due to itself will also be zero.

Step 3:
The principle of superposition of electric fields or forces states that the net electric field or force at any point due to a system of charges is the vector sum of the individual electric fields or forces at that point due to each charge in the system.

Therefore, the net electric field and force at the centre of the line joining the two charges Q due to the system of three charges will be zero if and only if the charge q is placed such that the electric field and force due to q itself is equal and opposite to the electric field and force due to the two charges Q.

Step 4:
Let us assume that the charge q is equal to -Q/4. Then, the electric field at the centre of the line joining the two charges Q due to q will be

E = kq/r^2 = k(-Q/4)/r^2 = -kQ/(4r^2)

where k is the Coulomb constant and r is the distance between q and each of the charges Q.

Similarly, the electric field at the centre of the line joining the two charges Q due to each of the charges Q will be

E = kQ/r^2

Therefore, the net electric field at the centre of the line joining the two charges Q due to the system of three charges will be

E(net) = -kQ/(4r^2) + kQ/r^2 = 0

Step 5:
Similarly, the net force at the centre of the line joining the two charges Q due to the system of three charges will be zero if and only if the force due to q itself is equal and opposite to the force due to the two charges Q.

The force on q due to each of the charges Q will be

F = kqQ/r^2 = k(-Q/4)Q/r^2 = -kQ^2/(4r^2)

Therefore, the net force on q due to the two charges Q will be

F(net) = -kQ^2/(4r^2) + kQ^2/r^2 = 0

Step 6:
Thus, we have shown that if the charge q is equal to -Q/4, then the system of three charges will be in equilibrium.

Therefore, the correct answer