b
(b)

In the string elastic force is conservative in nature.

\(\therefore W =-\Delta U\)

Work done by elastic force of string,

\(W =-\left( U _{ F }- U _{ i }\right)= U _{ i }- U _{ F }\)

\(W = \frac{1}{2} kx ^2-\frac{ k }{2}( x + y )^2\)

\(=\frac{1}{2} kx ^2-\frac{1}{2} k \left( x ^2+ y ^2+2 xy \right)\)

\(=\frac{1}{2} kx ^2-\frac{1}{2} ky ^2-\frac{1}{2} kx ^2-\frac{1}{2} k (2 xy )\)

\(=- kxy -\frac{1}{2} ky ^2\)

Therefore, the work done against elastic force

\(W _{\text {external }}=- W =\frac{ ky }{2}(2 x + y )\)