Editing Parabola (section)

== In a Cartesian coordinate system ==
=== Axis of symmetry parallel to the ''y'' axis ===
[[File:Parabel-def-p-v.svg|thumb|Parabola with axis parallel to {{math|''y''}}-axis; {{math|''p''}} is the ''semi-latus rectum'']]
If one introduces [[Cartesian coordinates]], such that <math>F = (0, f),\ f > 0,</math> and the directrix has the equation <math>y = -f</math>, one obtains for a point <math>P = (x, y)</math> from <math>|PF|^2 = |Pl|^2</math> the equation <math>x^2 + (y - f)^2 = (y + f)^2</math>. Solving for <math>y</math> yields
<math display="block">y = \frac{1}{4f} x^2.</math>

This parabola is U-shaped (''opening to the top'').

The horizontal chord through the focus (see picture in opening section) is called the ''latus rectum''; one half of it is the ''[[Conic section#Conic parameters|semi-latus rectum]]''. The latus rectum is parallel to the directrix. The semi-latus rectum is designated by the letter <math>p</math>. From the picture one obtains 
<math display="block">p = 2f.</math>

The latus rectum is defined similarly for the other two conics – the ellipse and the hyperbola. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, <math>p</math> is the radius of the [[osculating circle]] at the vertex. For a parabola, the semi-latus rectum, <math>p</math>, is the distance of the focus from the directrix. Using the parameter <math>p</math>, the equation of the parabola can be rewritten as
<math display="block">x^2 = 2py.</math>

More generally, if the vertex is <math>V = (v_1, v_2)</math>, the focus <math>F = (v_1, v_2 + f)</math>, and the directrix <math>y = v_2 - f </math>, one obtains the equation
<math display="block">y = \frac{1}{4f} (x - v_1)^2 + v_2 = \frac{1}{4f} x^2 - \frac{v_1}{2f} x + \frac{v_1^2}{4f} + v_2.</math>

'''Remarks''':
# In the case of <math>f < 0</math> the parabola has a downward opening.
# The presumption that the ''axis is parallel to the y axis'' allows one to consider a parabola as the graph of a [[polynomial]] of degree 2, and conversely: the graph of an arbitrary polynomial of degree 2 is a parabola (see next section).
# If one exchanges <math>x</math> and <math>y</math>, one obtains equations of the form <math>y^2 = 2px</math>. These parabolas open to the left (if <math>p < 0</math>) or to the right (if <math>p > 0</math>).

=== General position ===
[[File:Parabel-abc.svg|thumb|Parabola: general position]]
If the focus is <math>F = (f_1, f_2)</math>, and the directrix <math>ax + by + c = 0</math>, then one obtains the equation
<math display="block">\frac{(ax + by + c)^2}{a^2 + b^2} = (x - f_1)^2 + (y - f_2)^2</math>

(the left side of the equation uses the [[Hesse normal form]] of a line to calculate the distance <math>|Pl|</math>).

For a [[parametric equation]] of a parabola in general position see {{slink||2=As the affine image of the unit parabola}}.

The [[implicit equation]] of a parabola is defined by an [[irreducible polynomial]] of degree two:
<math display="block">ax^2 + bxy + cy^2 + dx + ey + f = 0,</math>
such that <math>b^2 - 4ac = 0,</math> or, equivalently, such that <math>ax^2 + bxy + cy^2</math> is the square of a [[linear polynomial]].