What If Not?

I imagine (or I hope) most teachers of Geometry, Algebra 2 and above are familiar with the construction of a parabola. As illustrated in the mathlet below, we begin with a line which serves as the directrix and a point not on the line which is our focus. Place a point on the directrix (labeled as Drag Me!). Since each point on the desired parabola is equidistant from the focus and the directrix, we next construct the perpendicular bisector of the "Drag Me" point and the focus, knowing the equidistant point in question must lie on this line. This bisector happens to be tangent to the parabola (which provides an interesting way to find the equation of a tangent line). We next construct a perpendicular to the directrix at the "Drag Me" point. The point where this line intersects the perpendicular bisector is on the parabola. In fact, the locus of this point as you drag the "Drag Me" point along the directrix is the parabola. If you right-click (or ctrl-click) on this point, you can choose trace the point.


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This is where the Art of Problem Posing and the What If Not strategy comes into play.



I played around with the following idea a couple of years ago in one of my classes. I did this as a lark, not quite certain where it would lead. I am glad I posed the questions!

Applying the What If Not strategy to the parabola construction, I changed the construction question. What if the directrix used in the construction were not the directrix, but instead, the parabola? In other words, what if we mimicked the parabola construction using the parabola as our directrix? What would happen?

Let's begin with a parabola, its focus, and its directrix (we still need this line so we can change the direction the parabola opens). Place a point (Drag Me! of course) on the parabola. To mimic the construction, we are looking for a point that is equidistant from the point Drag Me and the focus. This point must lie on the perpendicular bisector of these two points, so we construct it. Continuing to mimic the construction, we next construct a perpendicular to the parabola at the point Drag Me. To do this, we construct a tangent at the point Drag Me, and then a perpendicular to this. The intersection of the perpendicular bisector and this perpendicular is the desired point equidistant from the parabola and the focus. The locus of this point is known as Tschirnhausen's Cubic.



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What else can you do with this? It is a nice exercise to derive a parametric equation of this cubic. What if you used some other shape, say, a circle for the directrix in this construction? Do you recognize this change in the construction as leading to an ellipse or a hyperbola? What if you repeated this construction using Tschirnhausen's Cubic as the directrix? It's never ending! Applying the What If Not strategy to a common problem is a great way to find something new hiding beneath the surface!