15. Let $x, y$ be real numbers. Then
$$
\max _{S x^{2}+4 y^{2}=10 x}\left(x^{2}+y^{2}\right)=
$$
$\qquad$
15. 4 .
$$
\begin{array}{l}
\text { Given } 5 x^{2}+4 y^{2}=10 x \\
\Rightarrow 4 y^{2}=10 x-5 x^{2} \geqslant 0 \\
\Rightarrow 0 \leqslant x \leqslant 2 .
\end{array}
$$

Then $4\left(x^{2}+y^{2}\right)=10 x-x^{2}$
$$
\begin{array}{l}
=25-(5-x)^{2} \leqslant 25-3^{2} \\
\Rightarrow x^{2}+y^{2} \leqslant 4 .
\end{array}
$$