20. Let $V$ be the vertex of the parabola, and $F$ be the focus of the parabola. If a point $A$ on the parabola satisfies $A F=20, A V=21$, then the sum of all possible values of the length of $F V$ is ( ).
(A) 13
(B) $\frac{40}{3}$
(C) $\frac{41}{3}$
(D) 14
(E) $\frac{43}{3}$
20. B.

Let the directrix of the parabola be $l$, the projection of point $F$ on the directrix be $P$, and the projection of point $A$ on the directrix be $Q$.

Then $A F=A Q, F P$ intersects the parabola at point $V, V 、 F$ have projections $Y 、 X$ on the line $A Q$.

Let $F V=V P=d$.
(1) If $F X=V Y$, as shown in Figure 10.
Figure 10

Then $\sqrt{F A^{2}-A X^{2}}=\sqrt{A V^{2}-A Y^{2}}$
$$
\begin{array}{l}
\Rightarrow 20^{2}-(2 d-20)^{2}=21^{2}-(20-d)^{2} \\
\Rightarrow 3 d^{2}-40 d+41=0 \\
\Rightarrow d=\frac{20 \pm \sqrt{277}}{3} \\
\Rightarrow d_{1}+d_{2}=\frac{40}{3} .
\end{array}
$$
(2) If $F X=V Y$, as shown in Figure 11.
Figure 11

Then $A F^{2}-A X^{2}=A V^{2}-A Y^{2}$
$$
\begin{array}{l}
\Rightarrow A F^{2}-(A Q-2 d)^{2}=A V^{2}-(A Q-d)^{2} \\
\Rightarrow 20^{2}-(20-2 d)^{2}=21^{2}-(20-d)^{2} \\
\Rightarrow 3 d^{2}-40 d+41=0 \\
\Rightarrow d_{1}+d_{2}=\frac{40}{3} .
\end{array}
$$