For a body performing simple harmonic motion, its potential energy is \(E_x\) at displacement \(x\) and \(\mathrm{E}_{\mathrm{y}}\) at displacement y from mean position. The potential energy \(\mathrm{E}_0\) at displacement \((\mathrm{x}+\mathrm{y})\) is

Potential energy at \(\mathrm{x}=\mathrm{E}_{\mathrm{x}}=\frac{1}{2} \mathrm{kx}^2\)
\(\therefore \quad \mathrm{x}=\sqrt{\frac{2 \mathrm{E}_{\mathrm{x}}}{\mathrm{k}}}\)
Similarly,
Potential energy at \(\mathrm{y}=\mathrm{E}_{\mathrm{y}}=\frac{1}{2} \mathrm{ky}^2\)
\(\therefore \quad y=\sqrt{\frac{2 E_y}{k}}\)
P.E. at displacement \((\mathrm{x}+\mathrm{y})=\mathrm{E}=\frac{1}{2} \mathrm{k}(\mathrm{x}+\mathrm{y})^2\)
\(\begin{array}{ll}
\therefore & E=\frac{1}{2} k\left(x^2+y^2+2 x y\right) \\
\therefore & E=\frac{k}{2}\left[\frac{2 E_1}{k}+\frac{2 E_2}{k}+\frac{2 \times 2 \sqrt{E_1 E_2}}{k}\right] \\
\therefore & E=E_x+E_y+2 \sqrt{E_x E_y}
\end{array}\)