A designer draws a parabola whose focus is the point (2, 1) and whose directrix is the line y = -3…

Step 1. The distance. The focus is at height $1$ and the directrix at height $-3$, so they are $4$ units apart. This is $2a$, so $a = 2$.

Step 2. The vertex. It is halfway between: $\dfrac{1 + (-3)}{2} = -1$. The vertex is $(2, -1)$.

Step 3. The latus rectum. $4a = 8$.

Step 4. The equation. $(x - 2)^2 + (y - 1)^2 = (y + 3)^2$ gives $(x - 2)^2 = 8(y + 1)$.