# 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.
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 .

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