Conic Sections
Terms and Formulae
Terms
Axis
-
The line over which a parabola is symmetric.
Branch
-
The term for each of the two distinct sections of the graph of a hyperbola.
Center
-
For an ellipse and hyperbola, the midpoint between the foci. For a
circle, the fixed point from which all points on the circle are equidistant.
Circle
-
The set of all points equidistant from a given fixed point.
Conic
-
The intersection of a plane and a right circular cone.
Conjugate Axis
-
The line segment related to a hyperbola of length
2b
whose midpoint is the
center.
Degenerate Conic
-
A conic which is not a parabola, ellipse, circle, or
hyperbola. These include lines, intersecting lines, and points.
Diameter
-
A line segment that contains the center of a circle whose endpoints are
both on the circle, or sometimes, the length of that segment.
Directrix
-
For a parabola, it is the line whose distance from any point on the parabola
is the same as the distance from that point to the focus. For a conic
defined in polar terms, it is the line whose distance from any point on the
conic makes a constant ratio with the distance between that point and the focus.
Eccentricity
-
The ratio
in an ellipse or hyperbola. Under the polar
definition of conics,
e
is the constant ratio of the distance from a point
to the focus and the distance from that point to the directrix.
in an ellipse or hyperbola. Under the polar
definition of conics,
e
is the constant ratio of the distance from a point
to the focus and the distance from that point to the directrix.
Ellipse
-
The set of all points such that the sum of the distances from the point to each
of two fixed points is constant.
Focus
-
For a parabola, the point whose distance from any point on the parabola is
the same as the distance between that point and the directrix. For an
ellipse, one of two points--the sum of whose distances to a point on the
ellipse is constant. For a hyperbola, one of two points--the difference of
whose distances to a point on the hyperbola is constant. Under the polar
definition of a conic, it is the point whose distance from a point on the
conic makes a constant ratio with the distance between that point and the
directrix.
Hyperbola
-
The set of all points such that the difference of the distances between each of
two fixed points and any point on the hyperbola is constant.
Major Axis
-
The line segment containing the foci of an ellipse whose endpoints are
the vertices whose length is
2a
.
Minor Axis
-
The line segment containing the center of an ellipse perpendicular to
the major axis whose length is
2b
.
Parabola
-
The set of all points such that the distance between a point on the parabola and
a fixed line is the same as the distance between a point on the parabola and a
fixed point.
Radius
-
A segment between the center of a circle and a point on the circle, or
sometimes, the length of that segment.
Transverse Axis
-
The line segment that contains the center and whose endpoints are the two
vertices of a hyperbola.
Vertex
-
(Plural = "vertices") For a parabola, the point halfway between the focus and the
directrix. For an ellipse, one of two points where the line that
contains the foci intersects the ellipse. For a hyperbola, one of two
points at which the line containing the foci intersects the hyperbola.
Formulae
| Polar Form of a Conic |
r =  , orr = 
, where
e
is the eccentricity of the conic, the
pole is the focus,
and
p
is the distance between the focus and the directrix.
|
| Standard Form of a Circle | The standard equation for a circle is (x - h)2 + (y - k)2 = r 2 . The center is at (h, k) . The radius is r . |
| Standard Form of an Ellipse |
The standard equation of an ellipse with a horizontal major axis is the
following:
 +  = 1
. The center is at
(h, k)
. The length of the major axis is
2a
, and the length of the minor
axis is
2b
. The distance between the center and either focus is
c
,
where
c
2 = a
2 - b
2
.
a > b > 0
. The standard equation of an ellipse with
a vertical major axis is the following:
 +  = 1
. The center is at
(h, k)
. The length of the major axis is
2a
, and the length of the minor axis is
2b
. The distance between the center
and either focus is
c
, where
c
2 = a
2 - b
2
.
a > b > 0
.
|
| Standard Form of a Hyperbola |
The standard equation for a hyperbola with a horizontal transverse axis
is
- = 1
. The center is at
(h, k)
. The distance between the vertices is
2a
. The distance between
the foci is
2c
.
c
2 = a
2 + b
2
. The standard equation for a hyperbola
with a vertical transverse axis is
- = 1
. The center is at
(h, k)
. The distance between the vertices
is
2a
. The distance between the foci is
2c
.
c
2 = a
2 + b
2
.
|
| Standard Form of a Parabola | If a parabola has a vertical axis, the standard form of the equation of the parabola is this: (x - h)2 = 4p(y - k) , where p≠ 0 . The vertex of this parabola is at (h, k) . The focus is at (h, k + p) . The directrix is the line y = k - p . The axis is the line x = h . If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y - k)2 = 4p(x - h) , where p≠ 0 . The vertex of this parabola is at (h, k) . The focus is at (h + p, k) . The directrix is the line x = h - p . The axis is the line y = k . |
