q3 must be closer to q1 than to q2 for the force magnitudes to be the same, simply because q2 has 4 times the charge as q1. Therefore, B and D are eliminated right off.
The question acts about the MAGNITUDE of the force that q1 and q2 exert on q3; we don't have to worry about the direction of the force. The magnitudes are given by q1q3/(r13)^2 and q2q3/(r23)^2, but we know that q1q3 is 1/4 what q2q3 is. Therefore, r23 must be twice what r13 is, to make these magnitudes the same. In math-speak:
q*q/(r13)^2 = q*4q/(r23)^2, so that 1/(4r13^2) = 4/(4r23^2)
==> r23^2 = 4r13^2
==> r23 = 2*r13.
We have two possibilities: q3 is between q1 and q2, or (reading left-to-right) their positions are q3, q1, q2.
We know that r12 = 2.00 cm. Also, r13 + r23 = r12, so that r13 + 2*r13 = 2 cm, and so r13 = 0.66 cm. Here, q3 is 1/3 of the way between q1 and q2.
q3 to the left:
We know that r12 = 2.00 cm. Also, r13 + r12 = r23, so that r13 + 2 cm = 2 r13. Thus, r13 = 2cm (the same distance as r12).
From the diagrams given, if they're anything to scale, the only possible answer is (A), as in (C), r13 is obviously quite a bit less than r12. If they're NOT drawn to scale, then both (A) and (C) could be answers.