A particle is executing simple harmonic motion ( SHM ) of amplitude...

$\begin{cases}PE = \dfrac{1}{2}k{x}^{2} & \text{ and } KE = \dfrac{1}{2}K\left({A}^{2} - {x}^{2}\right)\end{cases}$

As given in question

$KE = PE$

$⇒ \dfrac{1}{2}K\left({A}^{2} - {x}^{2}\right) = \dfrac{1}{2}K{x}^{2}$

$⇒ 2{x}^{2} = {A}^{2}$

$⇒ x = \pm \dfrac{A}{\sqrt{2}}$