A body is executing S.H.M. At a displacement x its potential energy is 9   J and at a displacement y…

Consider an oscillating spring-mass system.

Potential energy at displacement $x$ will be written as, $\frac{1}{2}k{x}^2$.

Now,

$\frac{1}{2}k{x}^2=9 ...Eq\left(1\right)$

and

$\frac{1}{2}k{y}^2=16 ...Eq\left(2\right)$

Then, potential energy at the displacement is given by,

$$\begin{gathered} =\frac{1}{2}k{\left(x+y\right)}^2 \\ =\frac{1}{2}k\left(x^2+y^2+2xy\right) \\ =\frac{1}{2}k{x}^2+\frac{1}{2}k{y}^2+2\sqrt{\frac{1}{2}k{x}^2}\sqrt{\frac{1}{2}k{y}^2} \\ =9+16+2\times 3\times 4 \\ =49 \mathrm{J} \end{gathered}$$