Copyright © 2011 Pearson Education, Inc. The Parabola Section 7.1 The Conic Sections

7.1 Copyright © 2011 Pearson Education, Inc. Slide 7-3 The parabola, circle, ellipse, and hyperbola can be defined as the four curves that are obtained by intersecting a double right circular cone and a plane. That is why these curves are known as conic sections. If the plane passes through the vertex of the cone, then the intersection of the cone and the plane is called a degenerate conic. There are geometric definitions from which we derive equations of the conic sections. Definitions

7.1 Copyright © 2011 Pearson Education, Inc. Slide 7-4 A parabola is the set of all points in the plane that are equidistant from a fixed line (the directrix) and a fixed point not on the line (the focus). Definition: Parabola

7.1 Copyright © 2011 Pearson Education, Inc. Slide 7-5 In terms of the directrix and focus, the axis of symmetry can be described as the line perpendicular to the directrix and containing the focus. The vertex is the point on the axis of symmetry that is equidistant from the focus and directrix. If we use the coordinates (h, k) for the vertex, then the focus is (h, k + p) and the directrix is y = k – p, where p (the focal length) is the directed distance from the vertex to the focus. If the focus is above the vertex, then p > 0, and if the focus is below the vertex, then p < 0. The distance from the vertex to the focus or from the vertex to the directrix is |p|. The Parabola

7.1 Copyright © 2011 Pearson Education, Inc. Slide 7-6 The distance d 1 from an arbitrary point (x, y) on the parabola to the directrix is the distance from (x, y) to (x, k – p). We use the distance formula to find d 1 : Now we find the distance d 2 between (x, y) and the focus (h, k + p): Since d 1 = d 2 for every point (x, y) on the parabola, we have the following equation. Verify that squaring each side and simplifying yields The Parabola

7.1 Copyright © 2011 Pearson Education, Inc. Slide 7-7 The equation of a parabola with focus (h, k + p) and directrix y = k – p is y = a(x – h) 2 + k, where a = 1/(4p) and (h, k) is the vertex. Theorem: The Equation of a Parabola

7.1 Copyright © 2011 Pearson Education, Inc. Slide 7-8 The link between the geometric definition and the equation of a parabola is a = 1/(4p). For any particular parabola, a and p have the same sign. If they are both positive, the parabola opens upward and the focus is above the directrix. If they are both negative, the parabola opens downward and the focus is below the directrix. Since a is inversely proportional to p, smaller values of |p| correspond to larger values of |a| and to “narrower” parabolas. The Parabola

7.1 Copyright © 2011 Pearson Education, Inc. Slide 7-9 The graphs of y = 2x 2 and x = 2y 2 are both parabolas. Interchanging the variables simply changes the roles of the x- and y-axes. The parabola y = 2x 2 opens upward, whereas the parabola x = 2y 2 opens to the right. For parabolas opening right or left, the directrix is a vertical line. If the focus is to the right of the directrix, then the parabola opens to the right. If the focus is to the left of the directrix, then the parabola opens to the left. Parabolas Opening to the Left or Right