c
(c) For motion of the particle from \((0, 0)\) to \((a, 0)\)

\(\overrightarrow F = - K(0\,\hat i + a\,\hat j)\) \( \Rightarrow \,\,\overrightarrow F = - Ka\hat j\)

Displacement \(\overrightarrow {r\,} = (a\,\hat i + 0\,\hat j) - (0\,\hat i + 0\,\hat j) = a\hat i\)

So work done from \((0, 0)\) to \((a, 0)\) is given by

\(W = \overrightarrow F \,.\,\overrightarrow {r\,} \)\( = - Ka\hat j\,.\,a\hat i = 0\)

For motion \((a, 0)\) to \((a, a)\)

\(\overrightarrow F = - K(a\hat i + a\hat j)\) and displacement

\(\overrightarrow {r\,} = (a\hat i + a\hat j) - (a\hat i + 0\hat j) = a\hat j\)

So work done from \((a, 0)\) to \((a, a)\) \(W = \overrightarrow F \,.\,\overrightarrow {r\,} \)

\( = - K(a\hat i + a\hat j)\,.\,a\hat j = - K{a^2}\)

So total work done\( = - K{a^2}\)