If the sum of two positive quantities is a constant,
then their product is maximum, when they are equal.
$\therefore a^{2} x^{4} \cdot b^{2} y^{2}$ is maximum when
$a^{2} x^{4}=b^{2} y^{4}=\frac{1}{2}\left(a^{2} x^{4}=\frac{1}{2}\left(a^{2} x^{4}+b^{2} y^{4}\right)\right)$
$=\frac{c^{4}}{2}$
$\therefore $ maximum value of
$a^{2} x^{4} \cdot b^{2} y^{4}=\frac{c^{4}}{2} \cdot \frac{c^{4}}{2}=\frac{c^{8}}{4}$
Maximum value of
$x y=\left(\frac{c^{8}}{4 a^{2} b^{2}}\right)^{1 / 4}=\frac{c^{2}}{\sqrt{2 a b}}$