Equations for Conic Sections
The
general form of the equation for any conic section is:
A
x²+B
xy+C
y²+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±
ecos(θ) : vertical
directrix
or
r=(
ep)/(1±
esin(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: