The Parabola

1. Core Concept Overview & Intuition

A parabola is a conic section defined as a locus. It is the set of all points that move so that the ratio of their distance from a fixed point to their distance from a fixed line is exactly one. The fixed point is called the focus, denoted $F$, and the fixed line is called the directrix, denoted $DM$. Because the two distances are equal, every point on the parabola is equidistant from the focus and the directrix. This single defining property generates the entire curve and all of its standard equations.

The line passing through the focus and perpendicular to the directrix is called the axis of the parabola. The point where the axis meets the curve is the vertex, denoted $V$. The vertex is the midpoint of the perpendicular segment from the focus to the directrix, so it lies exactly halfway between them. The curve is symmetric about its axis, which is why the axis is also called the axis of symmetry. A parabola opens away from the directrix and toward the focus, so the relative position of the focus and directrix determines the orientation of the curve.

Two measurements control the size and shape of a parabola. The first is the distance from the vertex to the focus, conventionally denoted $a$. The second is the distance from the vertex to the directrix, which is also $a$ because the vertex is the midpoint between focus and directrix. The focus therefore lies $a$ units from the vertex along the axis, and the directrix is the line $a$ units from the vertex on the opposite side. A chord is a line segment whose endpoints lie on the parabola. A chord that passes through the focus is called a focal chord, and the focal chord perpendicular to the axis is called the latus rectum. The latus rectum is the natural "width" of the parabola measured through the focus, and its length is $4a$.

The parabola can be positioned in two broad ways. In the standard forms, the vertex is at the origin and the axis coincides with one of the coordinate axes. In the translated forms, the vertex is moved to an arbitrary point $(h,k)$ while the axis remains parallel to a coordinate axis. The same geometric definition and the same distance relationship govern both cases; only the coordinates of the vertex, focus, and directrix change. Understanding the standard case thoroughly makes the translated case a straightforward shift of coordinates.

Beyond its shape, the parabola supports a rich set of line relationships. A tangent is a line that touches the parabola at exactly one point, and its slope can be obtained by considering a second point on the curve and taking a limit as that point approaches the point of contact. A normal is the line perpendicular to the tangent at the point of contact, so its slope is the negative reciprocal of the tangent slope. These two lines, together with the focus, directrix, vertex, axis, and latus rectum, form the complete geometric toolkit for the parabola.

2. Mathematical Formalization & Core Definitions

Definition (Parabola). A parabola is the locus of a point which moves such that the ratio of its distance from a fixed point to the distance from a fixed line is one. The fixed point is the focus $F$, and the fixed line is the directrix $DM$.

Axis, vertex, and symmetry. The line passing through the focus and perpendicular to the directrix is the axis of the parabola. The point $V$ where the axis meets the curve is the vertex. The vertex is the midpoint of the perpendicular from the focus to the directrix, and the curve is symmetric about the axis.

Focal chord and latus rectum. A chord is a line segment whose endpoints are points on the parabola. A focal chord is a chord passing through the focus $F$. The focal chord perpendicular to the axis is the latus rectum. The distance of any point $P(x,y)$ on the curve to the focus is called its focal distance.

Standard derivation with vertex at the origin. Suppose the axis coincides with the $x$-axis and the vertex $V$ is at the origin. Let the distance from the vertex to the focus be $a$. Then the focus is $F(a,0)$. For any point $P(x,y)$ on the parabola, the definition requires $PF=PM$, where $M$ is the foot of the perpendicular from $P$ to the directrix. The directrix is the line $x=-a$, so $PM=x+a$ and $PF=\sqrt{(x-a)^2+y^2}$. Equating the two distances and squaring gives

$$
(x-a)^2+y^2=(x+a)^2.
$$

Expanding both sides and simplifying yields the standard equation

$$
y^2=4ax.
$$

This is the standard form for a parabola with vertex at the origin, axis along the $x$-axis, and opening in the positive $x$-direction. The focus is $F(a,0)$ and the directrix is $x=-a$.

General equation from the distance definition. The distance from a point $(x,y)$ to a line $Ax+By+C=0$ is used to convert the focus–directrix definition into a general equation. The general equation of the parabola is therefore given by the condition that the distance to the focus equals the distance to the directrix. For an axis parallel to the $x$-axis in the positive orientation the standard form is $y^2=4ax$; for an axis parallel to the $x$-axis in the negative orientation the corresponding form is obtained by the same distance condition with the focus placed on the negative side.

Translated parabolas with vertex $(h,k)$. When the vertex is moved to $(h,k)$ while the axis remains parallel to a coordinate axis, the standard forms become

$$
(y-k)^2=4a(x-h)
$$

for an axis parallel to the $x$-axis, and

$$
(x-h)^2=4a(y-k)
$$

for an axis parallel to the $y$-axis. In the first case the axis of symmetry is the horizontal line $y=k$, the vertex is $V(h,k)$, and the focus is $F(h+a,k)$. In the second case the axis of symmetry is the vertical line $x=h$, the vertex is $V(h,k)$, and the focus is $F(h,k+a)$. The directrix is correspondingly $x=h-a$ or $y=k-a$. The parameter $a$ retains its meaning as the distance from the vertex to the focus.

Parametric description. The standard parabola $y^2=4ax$ admits the parametric representation in which the coordinates of a general point are expressed in terms of a single parameter. The parametric equations are

$$
x=at^2,\qquad y=2at.
$$

Substituting these into $y^2=4ax$ verifies the representation, since $(2at)^2=4a(at^2)$. The parameter $t$ labels each point of the parabola, and the geometric properties of the curve can be expressed in terms of $t$.