A Degenerate Conic Section

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This applet illustrates an important case of a degenerate conic. Whenever we have three collinear points and two other points not on that line, then the equation of the conic is always uniquely determined, up to a multiplicative factor. Furthermore, it is an equation that can be factored into a product of two linear equations, one of which is satisfied by the three collinear points, the other of which is satisfied by the two additional points.

In the applet, you may click and drag the blue points A and B. (The red points are fixed.) The (degenerate) conic section passing through these points will be drawn. Note that if the points are arranged so that four or five are collinear, then there are infinitely many degenerate conic sections passing through these points, and no conic section will be drawn.

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Bradley and Stemkoski, Created with GeoGebra