Introduction:
In order to find the relation between the charges Q and q that will result in a net force of zero on q, we need to analyze the forces acting on q due to the other charges.

Analysis:
Let's consider the charges one by one and calculate the forces they exert on q. We can use Coulomb's Law to determine the force between two charges.

1. Force due to charge 2q:
The force exerted by charge 2q on q can be calculated using Coulomb's Law:
F1 = k * (2q) * q / r1^2
Where k is the electrostatic constant and r1 is the distance between charge 2q and q. Since charge 2q is located at point 0, the distance r1 is simply a.

2. Force due to charge Q:
The force exerted by charge Q on q can be calculated using Coulomb's Law:
F2 = k * Q * q / r2^2
Where r2 is the distance between charge Q and q. Since charge Q is located at point a/2, the distance r2 is (a - a/2) = a/2.

3. Force due to charge q:
The force exerted by charge q on itself is zero.

Net Force:
To find the net force on q, we need to consider the vector sum of the forces F1 and F2. Since the charges are placed along a straight line, the forces have opposite directions.

The net force on q can be calculated as:
Net Force = F1 - F2

For the net force to be zero, we need F1 to be equal to F2. Therefore, we have:
F1 = F2

Substituting the expressions for F1 and F2, we get:
k * (2q) * q / a^2 = k * Q * q / (a/2)^2

Simplifying the equation, we have:
4q^2 = Q * q

Conclusion:
The relation between Q and q that will result in a net force of zero on q is Q = 4q.