# Parabola

A **parabola** is a conic section generated by the intersection of a cone, and a plane tangent to the cone or parallel to some plane tangent to the cone. If the plane is itself tangent to the cone, one would obtain a degenerate parabola, a line. In other words, a parabola is the locus of points which are equidistant from a given point (the **focus**) and a given line (the **directrix**).

Table of contents |

2 Derivation of the focus 3 Reflective property of the tangent 4 Constructing a parabola 5 External links |

## Definitions and overview

In Cartesian coordinates, a parabola with an axis parallel to the *y* axis with vertex (*h*, *k*), focus (*h*, *k* + *p*), and directrix *y* = *k* - *p* has the equation

A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution. See also parabolic reflector.

A particle or body in motion under the influence of a uniform gravitational field (for instance, a baseball flying through the air, neglecting air friction) follows a parabolic trajectory.

### Equations (Cartesian):

Vertical axis of symmetry:### Equations (parametric):

## Derivation of the focus

Given a parabola parallel to the *y*-axis with vertex (0,0) and with equation

*f*) -- the focus -- such that any point

*P*on the parabola will be equidistant from both the focus and a line perpendicular to the axis of symmetry of the parabola (the

*linea directrix*), in this case parallel to the

*x*axis. Since the vertex is one of the possible points P, it follows that the linea directrix passes through the point (0,-

*f*). So for any point

*P=(x,y)*, it will be equidistant from (0,

*f*) and (

*x*,-

*f*). It is desired to find the value of

*f*which has this property.

Let *F* denote the focus, and let *Q* denote the point at (*x*,-*f*). Line *FP* has the same length as line *QP*.

*x*from both sides (

^{2}*x*is generally not zero),

*p=f*and the equation for the parabola becomes

## Reflective property of the tangent

The tangent of the parabola described by equation (1) has slope

This line intersects the*y*-axis at the point (0,-

*y*) = (0, -

*a x*), and the

^{2}*x*-axis at the point (

*x/2*,0). Let this point be called

*G*. Point

*G*is also the midpoint of points

*F*and

*Q*:

*G*is the midpoint of line

*FG*, this means that

*P*is equidistant from both

*F*and

*Q*:

*GP*is equal to itself, therefore the triangles ΔFGP and ΔQGP are congruent.

If follows that the angles *FPG* and *GPQ* are equal. Line *QP* can be extended beyond *P* to some point *T*, and line *GP* can be extended beyond *P* to some point *R*. Then and are vertical, so they are equal (congruent). But is equal to . Therefore is equal to angle *FPQ*.

The line *RG* is tangent to the parabola at *P*, so any light beam bouncing off point *P\* will behave as if line *RG* were a mirror and it were bouncing off that mirror.

Let a light beam travel down the vertical line *TP* and bounce off from *P*. The beam's angle of inclination from the mirror is *RPT*, so when it bounces off, its angle of inclination must be equal to *RPT*. But has been shown to be equal to . Therefore the beam bounces off along the line *FP*: directly towards the focus.

Conclusion: Any light beam moving vertically downwards in the concavity of the parabola (parallel to the axis of symmetry) will bounce off the parabola moving directly towards the focus. (See parabolic reflector.)