If $x^{2}=8 x+y$ and $y^{2}=x+8 y$ with $x \neq y$, then the value of $x^{2}+y^{2}$ is
(A) 9
(B) 49
(C) 63
(D) 21
(E) 56
Since $x^{2}=8 x+y$ and $y^{2}=x+8 y$, then $x^{2}-y^{2}=(8 x+y)-(x+8 y)=7 x-7 y$.

Factoring both sides, we obtain $(x+y)(x-y)=7(x-y)$.

Since $x \neq y$, then $x-y \neq 0$, so we can divide both sides by $x-y$ to obtain $x+y=7$.

Since $x^{2}=8 x+y$ and $y^{2}=x+8 y$, then

$$
x^{2}+y^{2}=(8 x+y)+(x+8 y)=9 x+9 y=9(x+y)=9 \cdot 7=63
$$

ANswer: $(\mathrm{C})$