540 Hooked on Conics hyperbola determined by Jeff and Carl along with the southern branch of the hyperbola determined by Kai and Jeff, we see that there is exactly one point in common, and this is where Sasquatch must have been when it called. y Kai 6 5 4 3 2 1 Jeff 987654321 1 2 3 4 5 6 y Kai Carl x 1 2 3 4 5 6 6 5 4 3 2 1 Jeff 98765432 Sasquatch 2 3 4 5 6 Carl x 1 2 3 4 5 6 To determine the coordinates of this point of intersection exactly, we would need techniques for solving systems of non-linear equations (which we won't see until Section 8.7), so we use the calculator4 Doing so, we get Sasquatch is approximately at (0.9629, 0.8113). Each of the conic sections we have studied in this chapter result from graphing equations of the form Ax2 + Cy 2 + Dx + Ey + F = 0 for different choices of A, C, D, E, and5 F . While we've seen examples6 demonstrate how to convert an equation from this general form to one of the standard forms, we close this chapter with some advice about which standard form to choose.7 Strategies for Identifying Conic Sections Suppose the graph of equation Ax2 + Cy 2 + Dx + Ey + F = 0 is a non-degenerate conic section.a If just one variable is squared, the graph is a parabola. Put the equation in the form of Equation 7.2 (if x is squared) or Equation 7.3 (if y is squared). If both variables are squared, look at the coefficients of x2 and y 2 , A and B. If A = B, the graph is a circle. Put the equation in the form of Equation 7.1. If A 6= B but A and B have the same sign, the graph is an ellipse. Put the equation in the form of Equation 7.4. If A and B have the different signs, the graph is a hyperbola. Put the equation in the form of either Equation 7.6 or Equation 7.7. a 4 That is, a parabola, circle, ellipse, or hyperbola - see Section 7.1. First solve each hyperbola for y, and choose the correct equation (branch) before proceeding. See Section 11.6 to see why we skip B. 6 Examples 7.2.3, 7.3.4, 7.4.3, and 7.5.3, in particular. 7 We formalize this in Exercise 34. 5