2021 AMC 12A Problem 20

Let $V = (0, 0),$ focus $F = (0, f),$ and directrix $y = -f,$ where $f = FV.$ A point $A = (x, y)$ on the parabola satisfies $x^2 = 4fy$ and $AF = y + f = 20,$ so $y = 20 - f.$ Also $AV^2 = x^2 + y^2 = 4fy + y^2$ $= 441.$ Substituting $y = 20 - f:$ $$\begin{aligned} &4f(20 - f) + (20 - f)^2 = 441 \\ &\;\Longrightarrow\; 3f^2 - 40f + 41 = 0. \end{aligned}$$ By Vieta’s formulas, the sum of the two possible values of $f$ is $\dfrac{40}{3}.$ Thus, the correct answer is B.