Pre-Calc -- Conic Sections

Definition of a Ellipse

The set of all points in a plane where the sum of the distances from a vertex to the two foci is constant for all vertices on the outside of the ellipse.

Definition of a Circle

The set of all points in a plane where the radius, r, from a point remains constant.

Definition of a Hyperbola

The set of all points in a plane where the difference of the distances from two fixed points, foci, are constant.

Circle Equation

(x-h)²+(y-k)²=r²

Major Axis

The axis where the vertices and foci of an ellipse lie.

Minor Axis

The axis where the co-vertices of an ellipse lie.

Equation of an Ellipse with a Horizontal Major Axis

(x-h)²/a²+(y-k)²/b²=1

Equation of an Ellipse with a Vertical Major Axis

(x-h)²/b²+(y-k)²/a²=1

Foci of an Ellipse Equation

c=+-SQRT(a²-b²)

Horizontal Hyperbola Equation

(x-h)²/a²-(y-k)²/b²=1

Vertical Hyperbola Equation

(y-k)²/a²+(x-h)²/b²=1

Asymptopes

Horizontal: y=b/ax; Vertical: y=a/bx

Hyperbola Foci

c=+-SQRT(a²+b²)

Definition of a Parabola

The set of all points on a plane such that the distance l between the focus (p) and directrix (-p) is equal to the distance m between any point (x,y) and the focus.

Equation for a Parabola with y-axis as A.O.S

x²=4py

Equation for a Parabola with x-axis as A.O.S

y²=4px

Focus of a Parabola

Horizontal: (p,0); Vertical: (0,p)