Conics

Conic sections are curves obtained by intersecting a cone with a plane. There are four primary types of conic sections:

Ellipse: A set of points where the sum of the distances to two fixed points (foci) is constant.
Parabola: A set of points equidistant from a fixed point (focus) and a line (directrix).
Hyperbola: A set of points where the difference of the distances to two fixed points (foci) is constant.
Circle: A special case of an ellipse where the two foci coincide at the center, and all points are equidistant from this center point.

Parabola

A parabola is defined as the set (locus) of points in a plane that are equidistant from a fixed point called the focus and a fixed line known as the directrix
The segment that passes through the focus parallel to the directrix and connecting two points on the parabola is called the latus rectum
equation is y = 1/4c (x - h)² + k
c = the distance from the vertex to the focus & directrix
(h,k) is the vertex
if it opens up, the focus is above the vertex
directrix is y = k + or - c
opens up or to the right if c > 0

Ellipse

An ellipse is defined is the set (locus) of points in a plane in which the sum of the distances from two fixed points (foci) to a point on the curve is a constant
(x - h/a)² + (y - k/b)² = 1 where a > b
center is (h, k)
c is the distance from the center of the ellipse to the foci
c²= a² - b²
foci = (h ± c, k) or (h, k ± c)
vertices = (h ± a, k)
covertices = (h, k ± b)
major axis length: 2a
minor axis length: 2b
directrix: h ± a/e
latus rectum: 2b²/a

Hyperbola

A hyperbola is defined as the set (locus) of points in the plane where the differences of distances to two fixed points (foci) is constant
(x - h/a)² - (y - k/b)² = 1
center is (h, k)
transverse axis: 2a
conjugate axis: 2b
vertices: (h ± a, k)
c² = a² + b²
foci = (h ± c, k)
latus rectum: 2b²/a

Circle

A circle is a simple closed curve in a plane that is equidistant from a fixed center point
(x - h)² + (y - k)²= r²
center is (h, k)
The radius is the distance from the center to any point on the circle
The diameter is the chord that passes through the center of the circle, equaling twice the radius
area = πr2
circumference = 2πr

Eccentricity