Conic Sections

Circle

the set of all points in a plane that are equidistant from a given point in the plane

equation of a circle

(x - h) squared + (y - k) squared = r squared

radius

the distance from the center to any point on the circle

Ellipse

the set of all points in the plane, the sum of whose distances from the two foci is constant

equation of an ellipse w/ horizontal major axis

(x - h) squared / a^2 + (y - k)^2 / b^2 = 1

equation of an ellipse w/ vertical major axis

(y - k) squared / a^2 + (x - h)^2 / b^2 = 1

a

length of semi major axis

b

length of semi minor axis

the endpoints of each axis are the

vertices

Eccentricity of an Ellipse

a measure that describes the shape of an ellipse, e = c/a

Value of E is close to 0

near center of an ellipse

Value of E is close to 1

far from center of an ellipse

value of b^2 using eccentricity

b^2 = a^2(1 - e^2)

Hyperbola

the set of all points in a plane in which the absolute value of the difference of the distances from the two foci is constant

center of a hyperbola

the midpoint of the line segment whose endpoints are the foci (c/2)

vertex of a hyperbola

point on each branch of the hyperbola that is nearest to the center

transverse axis

endpoints are the vertices, length is 2a

conjugate axis

perpendicular to transverse axis at center, 2b

standard form of equation of hyperbola

same an ellipse but one is subtracted from the other

Is a > b always in a hyperbola?

No

equations of asymptotes of hyperbolas

y - k = +- b/a(x-h)
or y - k = +-a/b (x - h)

equilateral hyperbola

hyperbola where a = b and has perpendicular asymptotes

rectangular hyperbola

equilateral hyperbola xy = c where asymptotes are x and y axis

parabola

set all of points that are the same distance from both a fixed point and a given line

vertex of parabola

halfway between focus and directrix

What is 4p / 1 over 4a

distance from vertex to focus or directrix

equations of parabolas

y = a(x - h)^2 + k or x = a(y - k)^2 + h

other equation of parabolas

(x - h)^2 = 4p(y - k) or (y - k)^2 = 4p(x - h)