Missing information in the text of the problem: q3=3 q1 = 3q
We can solve the problem by solving a system of two equations. In the first one, we write that the electrostatic force between charge 1 and 2 is equal to the force between charge 2 and 3. In the second equation, we write that the sum of the distances q1-q2 and q2-q3 is equal to 42 cm:
[tex]k_e \frac{q_1 q_2}{r_{12}^2}=k_e \frac{q_2 q_3}{r_{23}^2} [/tex]
[tex]r_{12}+r_{23}=0.42~m[/tex]
From the second equation we write
[tex]r_{12}=0.42-r_{23}[/tex]
While we can rewrite the first equation as
[tex] \frac{q}{r_{12}^2}= \frac{3q}{r_{23}^2} [/tex]
Substituting and solving, we find
[tex]r_{23}=0.99~m[/tex] or [tex]r_{23}=0.27~m[/tex]. With the first solution, [tex]r_{12}[/tex] would be negative, which doesn't make sense; therefore we must consider only the second solution, and we have:
[tex]r_{23}=0.27~m[/tex]
[tex]r_{12}=0.42-r_{23}=0.15~m[/tex]
