A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to :

From the question,

a = - Ax, where an is acceleration, x is displacement, and k is a proportionality constant, is the result of the information provided.

$$\begin{gathered} \frac{vdx}{dx}=-kx \\ \Rightarrow vdv=-kxdx \end{gathered}$$

Suppose that there is any displacement between 0 and x, and that the velocity

${\int }_{v_0}^{2}vdv=-{\int }_0^{x}kxdx$

$\frac{v^2-{v_0}^2}{2}=-\frac{k{x}^2}{2}$

$m\left(\frac{v^2-{v_0}^2}{2}\right)=-\frac{mk{x}^2}{2}$

$\Rightarrow ∆K\propto x^2$

Hence the correct answer is $x^2.$