Central Conics Left in Place

Abstract

A method is presented for creating a problem, solving it, and confirming that the solution is correct. The problem is to analyze a quadratic equation in two variables, in order to identify and draw, in its place – that is, without rotating the coordinate system – the central conic section that the equation defines. One creates the problem by choosing conjugate diameters for the conic that are not orthogonal; the solution requires finding orthogonal conjugate diameters, namely the axes. One can do this by the method of Apollonius, which is to intersect the conic with a concentric circle; the resulting four points are the vertices of a rectangle, whose sides are parallel to the axes. Along the way, by completing squares, one has found another pair of conjugate diameters, one of these being parallel to one of the axes of coordinates (or indeed to any line that one chooses). By sketching all three pairs of diameters, with their endpoints, one can see by inspection whether one’s computations are likely to have been correct. Those computations in turn serve to confirm the general formulas that are found, in terms of the coefficients of the equation, for the endpoints of the axes of an arbitrary central conic. © 2021 Uniwersytet Pedagogiczny. All rights reserved.