10. Let $C$ be a circle with radius $r$, centered at the point $(\sqrt{2}, \sqrt{3})$, where $r$ is a positive real number. When $x, y$ are both rational numbers, the point $(x, y)$ is called a rational point. Then the maximum number of rational points on the circle $C$ is ( ).
(A) $0_{R}$.
(B) 1 .
(C) 2 .
(D) 3 .
(E) infinitely many.
10. Let $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ be two distinct rational points on circle $C$, then $x_{1}^{2}-2 \sqrt{2} x_{1}+2+y_{1}^{2}$
$$
-2 \sqrt{3} y_{1}+3=x_{2}^{2}-2 \sqrt{2} x_{2}+2+y_{2}^{2}
$$
$-2 \sqrt{3} y_{2}+3$, i.e., $\left(x_{1}^{2}-x_{2}^{2}\right)+\left(y_{1}^{2}-y_{2}^{2}\right)$ $=2 \sqrt{2}\left(x_{1}-x_{2}\right)+2 \sqrt{3}\left(y_{1}-y_{2}\right)$. The left side of the equation is rational, while since $x_{1}-x_{2}$ and $y_{1}-y_{2}$ are not both zero, the right side is irrational, which is clearly impossible. Therefore, it is impossible to have two or more rational points, hence the answer should be (B) =