Problem:
Two point charges q and 4q are located at r1 and r2 respectively, on the xy plane. Find the magnitude Q and location R of a third charge to be placed on the plane such that the total force on each of the three charges vanishes.

Solution:
To find the magnitude Q and location R of the third charge, we need to consider the forces acting on each charge and set them equal to zero.

Step 1: Forces on charge q
The force on charge q due to the other charges can be calculated using Coulomb's law. The force between q and 4q is given by:

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

Where k is the electrostatic constant and r is the distance between the charges. Since the force on q should vanish, we set F1 equal to zero:

k * (q * 4q) / r^2 = 0

Simplifying this equation, we get:

q * 4q = 0

This implies that either q = 0 or q = -4q.

Step 2: Forces on charge 4q
Similarly, the force on charge 4q due to the other charges can be calculated using Coulomb's law. The force between 4q and q is given by:

F2 = k * (4q * q) / r^2

Setting F2 equal to zero, we have:

k * (4q * q) / r^2 = 0

Simplifying this equation, we get:

4q * q = 0

This implies that either q = 0 or q = -4q.

Step 3: Forces on the third charge Q
Now, let's consider the force on the third charge Q due to the other charges. The force between Q and q is given by:

F3 = k * (Q * q) / r1^2

The force between Q and 4q is given by:

F4 = k * (Q * 4q) / r2^2

Setting F3 + F4 equal to zero, we have:

k * (Q * q) / r1^2 + k * (Q * 4q) / r2^2 = 0

Simplifying this equation, we get:

Q * q / r1^2 + Q * 4q / r2^2 = 0

Dividing both sides of the equation by Q and q, we have:

1 / r1^2 + 4 / r2^2 = 0

Solving this equation for Q, we get:

Q = - q * (r1^2 / 4r2^2)

Conclusion:
In order for the total force on each of the three charges to vanish, the magnitude Q of the third charge should be equal to - q * (r1^2 / 4r2^2), where q is the charge of the first two charges. The location R of the third charge can be determined by the values of r1 and r2.