1) Let's put 
and 
both on the x-axis such that is on the right side of 
. The distance between the two charges is 
.
2) Let's put
on the x-axis and let's call 
the distance between charge 1 and charge 3. As a consequence, the distance between charge 2 and 3 will be 
: if r is positive, then charge 3 will be located on the left of charge 1, if r is negative, charge 3 will be located between charge 1 and charge 2.
3) The electrostatic force between charge 1 and charge 3 is
while the force between charge 2 and charge 3 is
4) The problem says the two forces are equal in intensity. Therefore we can write
. Using the equations written at step 3), and substituting 
and 
as mentioned in the problem, becomes
Using, we can solve the equation, and we get two solutions:
Both solutions are correct. With the former, the charge 3 is located 2 cm on the left of charge 1, while with the latter, charge 3 is located 0.67 cm on the right of charge 1, between charge 1 and 2.
2) Let's put
3) The electrostatic force between charge 1 and charge 3 is
while the force between charge 2 and charge 3 is
4) The problem says the two forces are equal in intensity. Therefore we can write
Using
Both solutions are correct. With the former, the charge 3 is located 2 cm on the left of charge 1, while with the latter, charge 3 is located 0.67 cm on the right of charge 1, between charge 1 and 2.