Mathematics · Ch 12 — Conic Sections

Standard Equation and Properties of a Parabola

A parabola is defined as the locus of a point $P$ that moves so that its distance from a fixed point, the focus $F$ , always equals its (perpendicular) distance from a fixed straight line, the directrix, which does not pass through $F$ . Unlike the circle's single-distance definition, this focus-directrix property involves two distances kept equal -- the defining idea behind all three remaining conics.

Deriving the standard equation $y^2=4ax$ . Take the focus at $F(a,0)$ , with $a>0$ , and the directrix as the vertical line $x=-a$ (so the origin, midway between them, lies on the curve). Let $P(x,y)$ be any point on the parabola. Its distance to the focus is

PF=\sqrt{(x-a)^2+y^2},

and its perpendicular distance to the directrix $x=-a$ is $|x+a|$ . By the defining property, $PF$ equals this distance:

\sqrt{(x-a)^2+y^2}=x+a

(taking $x+a\ge0$ , true for every point of this parabola, since it opens toward positive $x$ ). Squaring both sides,

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

Expanding both squares,

x^2-2ax+a^2+y^2=x^2+2ax+a^2.

The $x^2$ and $a^2$ terms cancel from both sides, leaving

y^2=4ax,

the standard equation of a parabola opening to the right, with vertex at the origin.

The other three orientations. Reflecting or rotating the same construction gives three further standard forms, distinguished only by which axis the parabola opens along and in which direction:

Equation	Focus	Directrix	Opens toward
$y^2=4ax$	$(a,0)$	$x=-a$	positive $x$
$y^2=-4ax$	$(-a,0)$	$x=a$	negative $x$
$x^2=4ay$	$(0,a)$	$y=-a$	positive $y$
$x^2=-4ay$	$(0,-a)$	$y=a$	negative $y$

(In every case $a>0$ ; the equation is chosen to match the point through which the curve is known to pass, so first determine the axis, then the sign, from the given data.) …

Figure 1Parabola: focus, directrix, vertex, axis and latus rectum

What this figure shows. A single open, U-shaped curve (parabola) is drawn opening to the right, symmetric about a horizontal axis line. The vertex of the curve sits at the origin where the axis line crosses it. A labelled point F sits on the axis to the right of the vertex, marked as the focus. A vertical straight dashed line is drawn to the left of the vertex, labelled as the directrix, with the perpendicular distance from a sample point on the curve to this line shown equal in length to the distance from that same point to F. A vertical chord is drawn passing through F, perpendicular to the axis, with its two endpoints on the curve, lab …

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