Parabolas as Geometric Loci

Definition Let a line, the generator, cross a fixed line, the axis, at a point $V$, the vertex of the cone, at an angle $\alpha$ with $0 < \alpha < \frac{\pi}{2}$. Turning the generator about the axis sweeps out a right circular double cone. A conic section is the curve in which a plane that does not pass through $V$ meets this cone. If $\beta$ is the angle between the plane and the axis, with $0 \le \beta \le \frac{\pi}{2}$, the curve is

A plane through $V$ gives a degenerate section instead, which is a single point, a single line or a pair of crossing lines.

Definition Let $F$ be a point and $L$ a line in the plane, with $F$ not on $L$. The parabola with focus $F$ and directrix $L$ is the set of all points $P$ in the plane whose distance to $F$ equals their distance to $L$,

where $\lvert PF \rvert$ is the distance $d(P, F)$ between the two points and $\operatorname{dist}(P, L)$ is the perpendicular distance from $P$ to $L$. Ellipses that are not circles and hyperbolas have a similar description, in which the distance to the focus is a fixed multiple $e$ of the distance to the directrix, with $0 < e < 1$ for an ellipse and $e > 1$ for a hyperbola. That multiple is the eccentricity, and the parabola is the case $e = 1$. This description returns with the polar equations of conics. The parabolas cut from a cone and the parabolas defined this way are the same curves.

Definition The directrix of a parabola is the fixed line $L$ in its definition. Distances to $L$ are measured perpendicular to $L$. For a horizontal line $y = c$ the distance from $(x, y)$ is $\lvert y - c \rvert$, and for a vertical line $x = c$ it is $\lvert x - c \rvert$. The parabola never meets its directrix and lies entirely on the same side of it as the focus. A point on $L$ has distance $0$ to $L$ but a positive distance to $F$. For a point $P$ on the other side, the segment from $P$ to $F$ crosses $L$ at a point $Q$, so $\operatorname{dist}(P, L) \le \lvert PQ \rvert < \lvert PF \rvert$.

Definition The focus of a parabola is the fixed point $F$ in its definition. It lies on the parabola’s axis of symmetry, the line through the focus perpendicular to the directrix, inside the curve, on the side toward which the parabola opens. Because every point $P$ of the parabola has $\lvert PF \rvert = \operatorname{dist}(P, L)$, the distance from $P$ to the focus, called its focal distance, can be read from the distance to the directrix without a square root.

Definition The axis of a parabola is the line through the focus perpendicular to the directrix, and the parabola is symmetric about it. The vertex is the point where the parabola meets its axis. It is the midpoint between the focus and the point where the axis meets the directrix. It is the point of the parabola closest to the focus and also the closest to the directrix, at the same distance from both. The distance from the vertex to the focus is written $\lvert p \rvert$ with $p \ne 0$, and the sign of $p$ records which way the parabola opens, as in the standard equations below.

Definition Let $p \ne 0$. The parabola with focus $(0, p)$ and directrix $y = -p$ has vertex $(0, 0)$ and equation

It opens upward if $p > 0$ and downward if $p < 0$. In the same way, the parabola with focus $(p, 0)$ and directrix $x = -p$ is $y^2 = 4px$, which opens right if $p > 0$ and left if $p < 0$. For the first one, the definition says

Both sides are nonnegative, so squaring gives the equivalent equation

which simplifies to $x^2 = 4py$. Moving the vertex to $(h, k)$ gives

The first has focus $(h, k + p)$ and directrix $y = k - p$. The second has focus $(h + p, k)$ and directrix $x = h - p$.

Definition The latus rectum of a parabola is the chord through the focus perpendicular to the axis. For $x^2 = 4py$ it lies on the line $y = p$, its endpoints are $(-2p, p)$ and $(2p, p)$, and its length is

For $y^2 = 4px$ the endpoints are $(p, -2p)$ and $(p, 2p)$. Each endpoint is $2\lvert p \rvert$ from the focus and also $2\lvert p \rvert$ from the directrix.