This fact sheet provides information about the key properties and equations of conic sections including parabolas, ellipses, circles, and hyperbolas. It defines their Cartesian and parametric forms, eccentricity, and conic discriminant. It also explains how to calculate the shortest distance from a point on the conic to its directrix and focus to determine the eccentricity.

1. FACT SHEET: CONIC SECTIONS
The Conic Equation
The Parabola
Cartesian Form:
Parametric Form:
Eccentricity:
Conic Discriminant:
The Ellipse
Cartesian Form:
Parametric Form:
Eccentricity:
Conic Discriminant:

2. The Circle
Cartesian Form:
Parametric Form:
Eccentricity:
Conic Discriminant:
The Hyperbola
Cartesian Form:
Parametric Form:
Eccentricity:
Conic Discriminant:
Conic Distances
Given the following:
P: A Point on the Conic Curve
F: Is the Focus
N: Is the Point on the Directrix which gives the
shortest distance to the point P

3. The first thing to note is that line PN is Perpendicular to the
Directrix.
The Eccentricity: Is the constant Ratio between PN, the shortest
line connecting the Point P to the Directrix, and PF, the line
connecting the Point P to the Focus.
The line PN can be calculated via the following:
Where are constants that define the Directrix, and
is the Point P.
The line PF can be calculated via the following:
We can find the Conic Equation of the Conic Section by equating
the two instances of :