Conic Sections

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## Conic Graph PaperA locus of points is a set of points satisfying some requirement. For instance, a circle with radius Given a line The line through the focus perpendicular to the directrix is called the If Conics can be sketched using "graph paper" as shown. The focus lies at the center of a sequence of concentric circles. Starting with a line through the focus, a family of parallel lines is constructed so that the line spacing is the same as the circle spacing. The axis is the line through the focus perpendicular to every line in this family. Begin by selecting one of the parallel lines as a directrix. Then select an intersection point on the axis between the focus and the directrix. This is our vertex. For instance, we might select the line 12 units to the left of the focus, and the point on the axis 5 units from the focus. The eccentricity of the conic through this point would be 5/7; the focal length would be 5. Since 5/7 < 1, we know that the conic will be an ellipse. Any grid point on the ellipse satisfies the following: if It's a bit like we're graphing a line of slope 5/7. We view 5/7 as rise over run, where "run" refers to line spacing and "rise" to circle spacing. Also graphed here are a parabola and one branch of a hyperbola. The hyperbola crosses the axis 4 units from the directrix and 8 units from the focus, so its eccentricity is 8/4 = 2/1 = 2. The "rise" is 2 and the "run" is 1, so, for every 1 line space we move to the right, we move 2 circle spaces out. The other branch of the hyperbola crosses 12 spaces to the right of the directrix. Finally, the parabola of course has eccentricity 1, so for every line space we move to the right, we move out 1 circle space. It's one thing to read this and another to do it yourself, so make sure to try it on paper. A print-ready sheet of graph paper is provided here for your convenience. If you take the directrix as the 12 |

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

Sul Ross State University Rio Grande College

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