If the mean and variance of eight numbers

Mean $=\bar{x}=\frac{3+7+9+12+13+20+x+y}{8}=10$

$\Rightarrow x+y=16$...(i)

Variance $=\sigma^{2}=\frac{\Sigma\left(x_{i}\right)^{2}}{8}-(\bar{x})^{2}=25$

$\sigma^{2}=\frac{9+49+81+144+169+400+x^{2}+y^{2}}{8}-100=25$

$\Rightarrow x^{2}+y^{2}=148$

From eqn. (i), $(x+y)^{2}=(16)^{2}$

$\Rightarrow x^{2}+y^{2}+2 x y=256$

Using eqn. (ii), $148+2 x y=256$

$\Rightarrow \quad x y=52$