The line of action of the resultant of two like parallel forces shifts by one-fourth of the distance between the forces when the two forces are interchanged. The ratio of the two forces is :

3 : 5

For \(\tau = 0\)

\(F_1 (L - x) = F_2 x\)

When forces are interchanged we have

\(F_2 (\frac{3L}{4} - x) = F_1 (x + \frac{L}{4})\)

Dividing the two equations

\(\Rightarrow \frac{L - x}{x + \frac{L}{4}} = \frac{x}{\frac{3L}{4} - x}\)

\(\frac{3L}{4} = x + \frac{3x}{4} + \frac{x}{4} = 2x\)

\(\Rightarrow 3L = 8x\)

Substituting the value of x :

\(F_1 \frac{5L}{8} = F_2 \frac{3L}{8}\)

\(\Rightarrow F_1 : F_2 = 3 : 5\)