Parabola from focus and directrix

Statement

From the locus definition, write the equation of the parabola with focus $F(2;0)$ and directrix $x = -2$.

Solution

For a generic point $P(x;y)$ we impose $d(P,F) = d(P,d)$, where the distance from the vertical directrix $x = -2$ is $|x + 2|$: $\sqrt{(x-2)^2 + y^2} = |x + 2|.$ Both sides are $\ge 0$: we square. $(x-2)^2 + y^2 = (x+2)^2 \implies x^2 - 4x + 4 + y^2 = x^2 + 4x + 4.$ The terms $x^2$ and $4$ cancel and what remains is $y^2 - 4x = 4x$, that is $\boxed{y^2 = 8x} \quad\Longleftrightarrow\quad x = \frac{y^2}{8}.$ This is a parabola with horizontal axis (the $x$-axis), vertex at the origin, opening to the right: consistent with the focus $F(2;0)$ to the right of the directrix $x = -2$.

Links

Topics: Geometric loci
Concepts: Directrix · Point-line distance · Focus · Parabola · Line
Skills: Analytical geometry · Solving equations
Exercise type: Geometric problem