Introduction:
In this scenario, we have a point charge Q located at the origin (0,0) and another charge q of the same magnitude but opposite sign located at point a,0. We need to find the position of a third point charge at point P, where the electrostatic forces due to Q and q will balance each other.

Analysis:
To determine the position of point P, we need to consider the electrostatic forces acting on it. The electrostatic force between two point charges is given by Coulomb's Law:

F = k * |q1 * q2| / r^2

Where F is the electrostatic force, k is the electrostatic constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.

Step 1: Calculating the force between Q and P:
Let's assume the charge at point P is q3. The electrostatic force between Q and P can be calculated as:

F1 = k * |Q * q3| / r1^2

where r1 is the distance between Q and P.

Step 2: Calculating the force between q and P:
The electrostatic force between q and P can be calculated as:

F2 = k * |q * q3| / r2^2

where r2 is the distance between q and P.

Step 3: Balancing the forces:
Since we want the electrostatic forces to balance each other at point P, we can equate F1 and F2:

k * |Q * q3| / r1^2 = k * |q * q3| / r2^2

Simplifying the equation, we get:

|Q * q3| / r1^2 = |q * q3| / r2^2

Since the magnitudes of the charges are the same, we can cancel them out:

|Q| / r1^2 = |q| / r2^2

Step 4: Solving for the position of P:
From the above equation, we can solve for r2 in terms of r1:

r2^2 = (|q| * r1^2) / |Q|

Taking the square root of both sides, we get:

r2 = sqrt((|q| * r1^2) / |Q|)

So, the position of point P will be at coordinates (r2, 0).

Conclusion:
By balancing the electrostatic forces between point charges Q and q, we have determined that the position of point charge P will be at coordinates (r2, 0), where r2 is given by the equation r2 = sqrt((|q| * r1^2) / |Q|).