### Hyperbolas, Ellipses, and Parabolas

In mathematics, a conic section is the intersection of a plane and a cone. As the angle and the location of the intersection is changed we produce different curves such as circles, ellipses, parabolas, and hyperbolas. The eccentricity is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular. The directrix of a conic section is the line which, together with the point known as the focus, serves to define a conic section as the locus of points whose distance from the focus is proportional to the horizontal distance from the directrix, with r being the constant of proportionality. If the ratio r = 1, the conic is a parabola, if r < 1, it is an ellipse, and if r > 1, it is a hyperbola. When the plane passes through the vertex of the cones the intersection can be a point, a line, or two lines depending on the angle of the intersection. This blog will discuss three particular conic sections: the hyperbola, ellipse, and parabola. The figure below illustrates each of these curves and shows how they are obtained with conic sections.

Hyperbolas:    A hyperbola resembles two mirrored parabolas and like ellipses have two foci and two vertices.

There are two standard forms of the hyperbola:

## Examples:

Ellipses:          Ellipses are figures that are shaped like an oval, they look like “squashed” circles. Circles are really just special cases of the ellipse.

The standard form of the ellipse is…

## Examples:

Parabolas:      A parabola is the graph of the quadratic function which has the general form

Parabolas are U-shaped curves and can open either upward or downward.

Here are some properties of parabolas:

## Examples:

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