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:

If you have any additional questions, feel free to drop by CAPS!  Check out our drop-in hours here, visit us on the 3rd floor of Zimmerman Library, visit the Online Learning Center , or call us at 277-7205 for more information about our services.


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