The correct option is A u2−v2a+b
Let x1 and x2 be the distances of the two positions from mean position.
Then with usual notations,
u2=ω2(A2−x21) ... (i)
v2=ω2(A2−x22) ... (ii)
a=ω2x1 ... (iii)
b=ω2x2 ... (iv)
Subtracting Eq. (ii) from Eq. (i),
u2−v2=ω2(x22−x21) ... (v)
Adding Eqs. (iii) and (iv),
a+b=ω2(x1+x2) ... (vi)
Since we want the shortest distance, we are looking for x2−x1.
Dividing Eq. (v) by Eq. (vi)
u2−v2a+b=x2−x1