Conic Sections

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## Conic DiametersIn a circle, a chord is called a diameter if it passes through the center of the circle. One way of characterizing the diameter is this. Given a particular chord, we can consider the family of parallel chords as shown in orange. The diameter for this family is the red chord that bisects every member of the family. The notion of a diameter being a chord that passes through the center doesn't apply to conics, because at this point we haven't even established that conics Consider the conic with focus
The Pythagorean Theorem implies that
On the other hand,
by a similar argument. Now, Now suppose that Notice that we showed more than we set out to prove. Not only do the points of the diameter lie on a single line, but that line is concurrent with the directrix and the line from the focus perpendicular to the family of parallel chords. So, to determine the diameter, we can first construct the perpendicular from The diameter of an ellipse consists of a single line segment passing through the center of the ellipse (which we have yet to establish). On a hyperbola there are two types of chords — those that connect the two branches and those that lie between two points on the same branch — and the diameters come in two types accordingly. If the family of chords connects the two branches, then the diameter consists of the entire line passing through the center of the hyperbola (shown here). If each chord lies in a single branch, then the diameter consists of two rays on the same line passing through the center, each pointing in an opposite direction. The case of the parabola will be considered next. |

**Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences**

Sul Ross State University Rio Grande College

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Copyright © 2013 Michael Luis Ortiz. All rights reserved.