Determining the shape of a conic section A conic section is a set { (r. y) c R' : ar' + bry + cy' + de +ey + f = 0} for some real numbers a, b, c, d, e and f, where at least one of a. b and e is not zero. The first step is to rewrite the equation as But we+ / =0, where v = B is a symmetric 2 x 2 matrix with non-negative trace, w is a 2 x I matrix and f is a number. (The trace of a square matrix is the sum of its diagonal entries.) If the linear system Be = w (with unknown being =) does not have a solution, then the conic section is a parabola. Otherwise, its shape can be determined using the following table (and any z with Be = w). B Shape of the set 30 The empty set Positive definite A point An ellipse Indefinite 0 A hyperbola =0 Two intersecting lines The empty set Not invertible One line