The mean of the numbers a, b, 8, 5, 10, is 6 and the variance is $6.80$. Then $a^2+b^2=$

We have,

$\text{ Mean }=6\Rightarrow \frac{a+b+8+5+10}{5}=6\Rightarrow a+b=7$ ... (i) 

and, Variance =6.80

$\begin{array}{l}\Rightarrow \frac{1}{5}\left(a^2+b^2+64+25+100\right)-6^2=6.80\\ \Rightarrow a^2+b^2+189-180=34\Rightarrow a^2+b^2=25 .. \left(ii\right)\end{array}$

Clearly, a= 3 and b = 4 satisfy these equations.