Wednesday, April 14, 2010

What If Not - Circle as Directrix

As another application of What If Not problem posing, let's take a look at the parabola construction. What if the directrix was not a line, but instead, what if the directrix were a circle?

As illustrated in the mathlet below, begin with a circle and a point (Focus) different from the center. Place a point on the circle (Drag Me). As we did with the parabola construction, we are looking for those points equidistant from the Focus and the point Drag Me. This equidistant point will lie on the perpendicular bisector of the Focus and Drag Me, so we construct that next. Following the lead of the parabola construction, we construct a perpendicular to the circle from the point Drag Me; this is simply a radius. The intersection of this radius and the perpendicular bisector is equidistant from the focus and the circle. The locus of this point as Drag Me moves along the circle is an ellipse.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

What is the next logical (at least in my mind) What If Not question to ask? What if the circle was not the directrix, but the ellipse was the directrix? What would happen then?

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