# Conic Sections

### Introduction to Conics

A conic is the intersection of a plane and a right circular cone. The four basic types of conics are parabolas, ellipses, circles, and hyperbolas. We've already discussed parabolas and circles in previous sections, but here we'll define them a new way. Study the figures below to see how a conic is geometrically defined.

Figure %: The four basic types of conics

In the conics above, the plane does not pass through the vertex of the cone. When the plane does intersect the vertex of the cone, the resulting conic is called a degenerate conic. Degenerate conics include a point, a line, and two intersecting lines.

The equation of every conic can be written in the following form: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 . This is the algebraic definition of a conic. Conics can be classified according to the coefficients of this equation.

The determinant of the equation is B 2 - 4AC . Assuming a conic is not degenerate, the following conditions hold true: If B 2 -4AC > 0 , the conic is a hyperbola. If B 2 -4AC < 0 , the conic is a circle, or an ellipse. If B 2 - 4AC = 0 , the conic is a parabola.

Another way to classify conics has to do with the product of A and C . Assuming a conic is not degenerate, the following conditions hold true: if AC > 0 , the conic is an ellipse or a circle. If AC < 0 , the conic is a hyperbola. If AC = 0 , and A and C are not both zero, the conic is a parabola. Finally, if A = C , the conic is a circle.

In the following sections we'll study the other forms in which the equations for certain conics can be written, and what each part of the equation means graphically.

## Take a Study Break

### Star Trek gets SEXY

Chris Pine and Zoe Saldana heat up the red carpet!

### Are you afraid of relationships?

Auntie SparkNotes can help!

### Sexy starlet style

See every single look from the Met Gala!

### Geeky Actors: Then and Now

Travel back in time!

### Villains We Want These Actresses to Play

From super cute to super bad!

### 10 Movies Better Than Their Books

What do you think?

### How To Look Like J-Law...

When you don't look like J-Law.

### 12 Scientific Inaccuracies in Into Darkness

What did Star Trek get wrong?

## The Book

### Read What You Love, Anywhere You Like

Get Our FREE NOOK Reading Apps