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Secondary Magazine - Issue 46: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 20 October 2009 by ncetm_administrator
Updated on 12 November 2009 by ncetm_administrator

Secondary Magazine Issue 46cones

Focus on...Conic Sections

The four conic sections are the circle, ellipse, parabola and hyperbola:
conic sections

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Both Euclid and Archimedes are known to have studied conic sections, but the first work to explore them is thought to have been carried out by another ancient Greek mathematician, Menaechmus (380–320 BC).

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Conic sections can be defined by the locus of a point moving according to certain rules. For a circle the point moves so that it is a constant distance from one other fixed point, but if it moves so that the sum of the distances from two fixed points is a constant then an ellipse is formed. A hyperbola is the locus of a point moving so that the difference of the distance from two fixed points is a constant.
A parabola is the locus of the point which moves so that its distance from a fixed point is the same as its perpendicular distance from a line.

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Conic sections can also be defined by the equation of the line on a Cartesian grid.

A circle is defined by the equation:

x^2  + y^2  = r^2

An ellipse is defined by:
 \frac{x^2 }{a^2 }  + \frac{y^2 }{b^2 }  = 1 

A hyperbola is defined by:
\frac{x^2 }{a^2 }  - \frac{y^2 }{b^2 }  = 1

And a parabola is defined by:
x^2  = 4ay

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The Cartesian definitions arise from the graph of the general quadratic equation Ax^2  + Bxy + Cy^2 + Dx + Ey + F = 0 which generates each of the conic sections depending on whether B^2  - 4AC is positive, negative or equal to zero (giving a hyperbola, an ellipse or a parabola respectively. If A = C and B = 0 then the equation defines a circle).
You can explore this equation further using this Geogebra applet

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In the 17th century, the German mathematician and astrologer Johannes Kepler realised that the orbits in which planets move around the sun are elliptical, rejecting the idea of circular orbits.

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The light cone created by a table lamp with a circular shade generates the conic sections on its surroundings. The circle can be seen on the ceiling above the lamp but, if the lamp is tilted, this becomes an ellipse.
The hyperbola can be seen on the walls directly above and below the lamp.
There are images and other demonstrations on The Garden of Archimedes website.

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Are McDonald’s golden arches made from two parabolas? Find out from the Texas A&M University Mathematics Department website.

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