21. A circle $\Gamma$ with radius $r(r \in[a, b))$ passes through the two foci of the ellipse $C: x^{2}+16 y^{2}=16$ and intersects the ellipse $C$ at four points. Then the value of $a+b$ is ( ).
(A) $5 \sqrt{2}+4$
(B) $\sqrt{17}+7$
(C) $6 \sqrt{2}+3$
(D) $\sqrt{15}+8$
(E) 12
21. D.

From the given information, the two foci of the ellipse $C$ are
$$
F_{1}(-\sqrt{15}, 0), F_{2}(\sqrt{15}, 0) \text {. }
$$

Since the circle $\Gamma$ with radius $r$ passes through the two foci of the ellipse $C$ and intersects the ellipse $C$ at four points, the center of the circle lies on the $y$-axis. When the radius is the smallest, the diameter of the circle $\Gamma$ is $2 \sqrt{15}$.
Thus, $r \geqslant \sqrt{15}$.
When the circle is tangent to the ellipse,
$$
r^{2}-(r-1)^{2}=15 \Rightarrow r=8 \text {. }
$$

Since the circle $\Gamma$ intersects the ellipse $C$ at four points, we know $r<8$.
Therefore, $a=\sqrt{15}, b=8, a+b=8+\sqrt{15}$.