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Exploration idea #497

Sunday 1 March 2015 I recently learned of the following interesting property for a parabola:

Two perpendicular lines that are both tangent to a parabola will intersect at a point on the directrix of the parabola.

Many students do not know what the directrix of a parabola is - which is a bit of a shame because then they are not aware of the fact that a parabola is the set of points (locus of points) that are equidistant from a line (directrix) and a point not on the line (focus).

Here is the outline of a way to start to explore (discover?) this property of all parabolas.

Choose a fairly simple quadratic equation such that the vertex of the parabola is the origin or at least on the y-axis. Let's consider the graph of . The derivative, , gives the slope of the tangent at any point on the parabola. Select a point at which to find the equation of the tangent. Let's choose the point where  The coordinates of the point on the parabola are and the slope of the tangent is . Thus, the equation of the tangent line at the point is . Now need to find the point on the parabola where the tangent line will be perpendicular to the tangent line we already have, i.e. .  Hence, need to find point where derivative is equal to . This occurs at . The point of tangency is and the equation of the tangent line is . The two perpendicular tangent lines, and , intersect at the point . These results are shown in the graph at left.

Let's repeat this procedure for another point on the parabola - choosing .  The equation of the tangent at is . The point on the parabola where derivative is equal to is - and the equation of the tangent at this point is .  The two perpendicular tangent lines intersect at . The results for this 2nd set of perpendicular tangent lines are shown below right.  Note that the point of intersection of the two perpendicular tangent lines has a y-coordinate of -1; same as for the first pair above. Since any two distinct points must be collinear, we must repeat the procedure for at least one more point on the parabola.

Choosing a starting point of and following the same procedure as before produces the two perpendicular tangent lines of and that intersect at .  Results for this 3rd set of perpendicular tangent lines is shown at left.

So, if the parabola property I mentioned at the start of this blog entry was unknown then someone would hopefully conjecture the property from these three results.

What about this being a nice topic for a Maths HL or Maths SL student Exploration (IA task)?  I think it could be a suitable topic. I think a key aspect that would make it a strong Exploration is for a student to prove this property. This would require some thoughtful input from the student on setting up parameters and variables - and then organizing a clear algebraic proof annotated with clearly written text and supported by clearly labeled diagrams.  I think it is also conducive for a good student to think of some nice extensions. For example, do any other conic sections (ellipse, hyperbola, etc) have a similar property?  Do other conic sections have a 'directrix'?  If so, what role does it play in constructing the particular conic section as a locus of points?  Is there a way to illustrate the property of parabolas shown above using another procedure?

And, finally, would it not be nice if students could submit their Exploration electronically so that it could include dynamic images. For this particular Exploration a very nice addition would be an animation showing how by dragging one point on the graph of the parabola would automatically show the two perpendicular tangent lines and their intersection point - and how a dynamic 'capture' of these intersection points would 'trace out' the directrix.