Let p be a real number. The parabola with focus at (0,p) and directrix y=-p is symmetric about the y-axis and has the equation . If p>0, then the parabola opens upward; if p<0, then the parabola opens downward.
The parabola with the focus at (p, 0) and directix x =-p is symmetric about the x-axis and has the equation . If p>0, then the parabola opens to the right; if p<0, then the prabola opens to the left.
Each of these parabolas has its vertex at the origin.

Equations of Standard Ellipses

An ellipse centered at the origin with foci and at ( and the vertices and at has the equation
, where
An ellipse centered at the origin with the foci at and vertices at has the equation
, where
In both cases, and , the length of the long axis (called the major axis) is 2a and the length of the short axis (called the minor axis) is 2b.

Equations of Standard Hyperbolas

A hyperbola centered at the origin with foci and at and vertices and at has the equation
, where
The hyperbola has asymptotes .
A hyperbola centered at the origin at the origin with foci at and vertices at has the equation
where
The hyperbola has asymptotes
In both cases, c>a>0 and c>b>0

Eccentricity- Directix Theorem

Suppose l is a line, F is a point on l, and e is a positive real number. Let C be the set of points P in a plane with the property that , where is the perpendicular distance from P to L
1) If e=1, C is a parabola
2) If 0<e<1, C is an ellipse
3) If e>1 , C is a hyperbola

Polar Equations of Conic Sections

Let d>0. The conic section with a focus at the origin and eccentricity e has the polar equation
or
The conic section section with a focus at the origin and eccentricity e has the polar equation
or
If 0<e<1, the conic section is an ellipse; if e -1, it is a parabola; and if e>1, it is a hyperbola. The curves are defined over any interval in of length