Ellipse
Ellipse, Parabola, Hyperbola
Ellipse with center C(x0, y0)
and major axis parallel to x axis
Length of major axis A'A — 2a
Length of minor axis B'B = 2b Distance from center C to focus
F or F' is
c = Ö(a2
– b2)
Eccentricity = ε = c/a = {Ö(a2
– b2)}/a
Equation in rectangular coordinates:
Equation in polar coordinates if C is at O:
Equation in polar coordinates if C is on x axis and F'
is at 0:
If P is any point on the ellipse, PF + PF' = 2a
If the major axis is parallel to the y axis, interchange x and
y in the above or replace θ by ½π - θ [or 90° - θ].
Parabola with axis parallel to x axis
If vertex is at A(x0,y0) and the distance from
A to focus F is a > 0, the equation of the parabola is if
parabola opens to right
(y — y0)2 = 4a(x — x0)
If parabola opens to left
(y — y0)2 = -4a(x — x0)
If focus is at the origin the equation in polar coordinates is
In case the axis is parallel to the y axis, interchange x and
y or replace θ by ½π — θ [or 90° — θ].
Hyperbola with center C(x0, y0)
and major axis parallel to x axis
Length of major axis A'A = 2a
Length of minor axis B'B = 2b
Distance from center C to focus F or F' = c =
Ö(a2 + b2)
Eccentricity ε = c/a = Ö(a2
+ b2)/a
Equation in rectangular coordinates:
Slopes of asymptotes G'H and GH' = ±b/a
Equation in polar coordinates if C is at O:
Equation in polar coordinates if C is on X axis and F'
is at O:
If P is any point on the hyperbola, PF - PF' = ±2a [depending
on branch]
If the major axis is parallel to the y axis, interchange x and
y in the above or replace θ by ½π - θ [or 90° - θ].
