Equations for Conic Sections

Rectangular Form

The general form of the equation for any conic section is:


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=xcos(θ)-ysin(θ)  and  y=xsin(θ)+ycos(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


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

Polar Form

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
  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: