I’m going to talk to my daughter’s 8th grade class about conic sections. I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry. But I found it surprisingly tricky to work out examples. Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4. Using Pythagoras’ theorem produces an equation with a bunch of square roots. Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms. Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2. Am I doing something wrong? Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.? Of course, I then want to talk about light waves bouncing from one focus to another… I’m not sure how to justify this without a little calculus. - Cris