Conic sections An example of an ellipse: 4x2 + 9y2 = 1. 0.3 0.2 0.1 -0.4 -0.2 0.2 0.4 -0.1 -0.2 -0.3 An example of a hyperbola: x2 - y2 = 1. 4 2 -1 1 2 -2 -4 An example of a parabola: y = x2. 2 1 -2 -1 1 2 -1 -2 1
Standard forms for conic sections are: · The ellipse: 2 .2 - 1 w : 1 where a > 0, b > 0. Often we also insist that a > b. · The parabola: y2 = 4ax where a > 0. · The hyperbola: @2 62 y2 - 2 = 1 where a > 0, b > 0. x To bring an equation into standard form, we need the formula for rotating axes by an angle 0: P C * y 0 1 * X y 1 0 X 0 x = x' cos 0 - y' sin 0 y = x' sin 0 + y cos 0 2