Geometric Reasoning for Reflections on Conic

From MathWiki

It is common in high school discussions of conics to:

  • introduce the conics as the locus of points with a certain distance constraint;
  • derive that the conics have quadratic equations;
  • then appear to respond to the question of what use these are by referring to applications that are based on special reflection properties for light or sound.

There is a significant gap between the first two properties - for which there is reasoning - and the last which appears without real reasoning, except for students who are studying calculus.

We propose an analysis which looks at constrained local change to 'see' that the locus constraints imply that the path (reduced to a vector) has the desired reflective properties.

Here are some sample GSP Sketches for this:

GSP Sketch for the case of the ellipse. Notice that the 'vector of change' reasoning shows the equality of angles made by the two rays from foci.

GSP Sketch for the case of the parabola.