Solution:

Given, two point charges 4q and -q are at a distance r.

Let their position be A and B respectively, and the midpoint of AB be C.

Now, a third charge Q is placed at point C.

We need to find the value of Q such that the resultant force acting on -q is zero.

To solve this problem, we can use the principle of superposition.

Principle of Superposition:

According to this principle, the net force on a point charge due to a system of charges is the vector sum of the individual forces exerted by each charge on the given point charge.

Let F1, F2, and F3 be the forces exerted on -q by 4q, -q, and Q respectively.

Since the net force on -q is zero, we have:

F1 + F2 + F3 = 0

Now, we know that the force between two point charges q1 and q2, separated by a distance r, is given by:

F = k (q1 q2) / r^2

where k is the Coulomb's constant.

Using this formula, we can find the magnitudes and directions of the forces F1, F2, and F3.

Magnitude and Direction of F1:

The distance between 4q and -q is r.

Therefore, the magnitude of the force exerted by 4q on -q is:

F1 = k (4q) (-q) / r^2

= -3kq^2 / r^2

The direction of this force is towards 4q.

Magnitude and Direction of F2:

The distance between -q and -q is 0.

Therefore, the magnitude of the force exerted by -q on itself is zero.

Magnitude and Direction of F3:

The distance between Q and -q is r/2.

Therefore, the magnitude of the force exerted by Q on -q is:

F3 = k (Q) (-q) / (r/2)^2

= -4kQq / r^2

The direction of this force is towards Q.

Substituting the magnitudes and directions of F1, F2, and F3 in the equation F1 + F2 + F3 = 0, we get:

-3kq^2 / r^2 - 4kQq / r^2 = 0

Simplifying this equation, we get:

Q = 3q / 4

Therefore, the value of Q is 3q/4.

Hence, the correct option is (A) q.

Note: The principle of superposition is a powerful tool in solving problems involving multiple point charges. By using this principle, we can find the net force on a given charge due to any number of charges in its vicinity.