# Conics

- Author:
- Zachry Engel

## Equations of Four Standard Parabolas

Let p be a real number. The parabola with focus at (0,p) and directrix y=-p is symmetric about the y-axis and has the equation . If p>0, then the parabola opens upward; if p<0, then the parabola opens downward.
The parabola with the focus at (p, 0) and directix x =-p is symmetric about the x-axis and has the equation . If p>0, then the parabola opens to the right; if p<0, then the prabola opens to the left.
Each of these parabolas has its vertex at the origin.

## Equations of Standard Ellipses

An ellipse centered at the origin with foci and at ( and the vertices and at has the equation
, where
An ellipse centered at the origin with the foci at and vertices at has the equation
, where
In both cases, and , the length of the long axis (called the major axis) is 2a and the length of the short axis (called the minor axis) is 2b.

## Equations of Standard Hyperbolas

A hyperbola centered at the origin with foci and at and vertices and at has the equation
, where
The hyperbola has asymptotes .
A hyperbola centered at the origin at the origin with foci at and vertices at has the equation
where
The hyperbola has asymptotes
In both cases, c>a>0 and c>b>0

## Eccentricity- Directix Theorem

Suppose l is a line, F is a point on l, and e is a positive real number. Let C be the set of points P in a plane with the property that , where is the perpendicular distance from P to L
1) If e=1, C is a parabola
2) If 0<e<1, C is an ellipse
3) If e>1 , C is a hyperbola

## Polar Equations of Conic Sections

Let d>0. The conic section with a focus at the origin and eccentricity e has the polar equation
or
The conic section section with a focus at the origin and eccentricity e has the polar equation
or
If 0<e<1, the conic section is an ellipse; if e -1, it is a parabola; and if e>1, it is a hyperbola. The curves are defined over any interval in of length