Parabola:

A parabola is a set of all points in a plane that are equidistant from a given fixed point (the Focus) and a given straight line (the Directrix).

Different cases of parabolas:

1) With the vertex at the origin, the parabola opens in the positive x direction and has the equation where vertex=(0,0) and focus is the point (p,0).
2) With the vertex at the origin, the parabola opens in the negative x direction and has the equation where vertex=(0,0) and focus is the point (p,0).
3) With the vertex at the origin, the parabola opens in the positive y direction and has the equation where vertex=(0,0) and focus is the point (0,p).

4) With the vertex at the origin, the parabola opens in the negative y direction and has the equation  where vertex=(0,0) and focus is the point (0,p).

                       
Definition of an ellipse:

An ellipse is a set of all points in a plane, whose distances from two fixed points (the foci) is a positive constant.

Different cases of ellipses:

1) The vertex is at the origin and the foci and the major axis are on the x-axis with the center at the origin and has the equation of the form where the foci and the major axis are on the x-axis, the length of the major axis is 2a, the minor  axis is on the y-axis, the length of minor axis equals to 2b and the center of the origin is at the origin (0,0).

 2) The vertex is at the origin and the foci and the major axis are on the y-axis with the center at the origin and has the equation of the form where the foci and the major axis are on the y-axis, the length of the major axis=2a, the minor axis is on the x-axis, length of the minor axis=2b and the center is at the origin (0,0).

Definition of a hyperbola:

A hyperbola is a set of all points in a plane, the difference of whose distances from two fixed points (the foci) is a positive constant.

Different cases of hyperbolas:

1) The center is at the origin and the foci are on the x-axis and conjugate axis is the y-axis and has the equation of the form where the foci and the vertices are on the

x-axis, the distance between the foci=2a, the conjugate axis is on the y-axis and the center is at the origin (0,0).

2) The center is at the origin and the foci are on the y-axis and conjugate axis is the x-axis and has the equation of the form  where the foci and the vertices are on the

y-axis, the distance between the foci=2a, the conjugate axis is on the x-axis and the center is at the origin.

 Asymptotic Equations:

The equations of the asymptotes to the hyperbola  are as follows and .

Flash Movie	Flash Game	Flash Card
Core Concept Tutorial	Problem Solving Drill	Review Cheat Sheet

This tutorial introduces the properties of the conic sections. The conic sections are presented with the use of graphical illustrations. Parabolas, ellipses and hyperbolas are presented in the examples.

The definition of ellipses and hyperbolas are presented through the use of graphs and the examples in this tutorial. The different types of hyperbolas and ellipses are presented with the use of graphs in this tutorial. The foci and vertices of hyperbolas and ellipses are used to construct the graphs of hyperbolas and ellipses.

Specific Tutorial Features:
• Step by step examples of the conic sections and their properties.
• “Completion of square” trick is used to rewrite the equations of the conic sections.

Series Features:
• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
• Definition slides introduce terms as they are needed.
• Visual representation of concepts
• Animated examples—worked out step by step
• A concise summary is given at the conclusion of the tutorial.

Conic Sections
Conic Sections and their origins
Some applications of conic sections
Parabola
Parabola and its definition
Different types of parabolas and their equations in standard form
Ellipse
Ellipse and its definition
The foci and axes of an ellipse
Different types of equations of an ellipse and their equations
in standard form
Hyperbola
Hyperbola and its definition
The foci and vertices of a hyperbola
Different types of equations of a hyperbola and their equations
in standard form
Hyperbola and their asymptotes