### Equations for Conic Sections

The

general form of the equation for any conic section is:

A

`x`²+B

`xy`+C

`y`²+D

`x`+E

`y`+F=0

where

`x`,

`y`, A, B, C, D, E, and F are all

real numbers. To find the type of

conic, use this:

B²-4AC>0 :

hyperbola (or its

degenerate case of 2

intersecting lines)

B²-4AC=0 :

parabola (or one of its degenerate cases: 2

parallel lines, a single

line, or

nothing)

B²-4AC<0 :

ellipse if A≠C or

circle if A=C (or the degenerate case of a

point or

nothing)

If a B

term is present, this indicates that the

axes of the

graph have been rotated, and need to be rotated back in order to make the equation easier to work with. This is done by eliminating the

`xy` term. First, use the equation

(A-C)/B =

cot(2θ)

to solve for θ, then use the equations

`x`=

`x`′

cos(θ)-

`y`′

sin(θ) and

`y`=

`x`′

sin(θ)+

`y`′

cos(θ)

to put the general form equation in terms of

`x`′ and

`y`′. This will eliminate the

`xy` term.

With the equation now in the form

A′

`x`′²+C′

`y`′²+D′

`x`′+E′

`y`′+F′=0

more information can be found by putting that equation into the standard form for each conic section.

The standard form of the equation of a conic section in

polar form with one

focus located and the

pole is

`r`=(

`ep`)/(1±

`e`cos(θ) : vertical

directrix
or

`r`=(

`ep`)/(1±

`e`sin(θ) : horizontal

directrix
where

`r`,

`e`, and

`p` are

real numbers and θ goes from [0,2π) or [0°,360°).

`e` is called the

eccentricity, and is used to find the type of conic like this:

`e`<1 :

ellipse (

`e`=0 is a

circle)

`e`=1 :

parabola
`e`>1 :

hyperbola
The

directrix of the conic will be

`p` units away from the focus, in the positive direction if the

denominator is added, and in the negative direction if the

denominator is subtracted.

For more information, see the writeups on the individual conic sections: