994 CHAPTER 10 Analytic Geometry Geometric Definition and Equations of Parabolas The equation of a parabola comes from the geometric definition of a parabola as a set of points. Parabola A parabola is the set of points in a plane equidistant from a fixed point and a fixed line. The fixed point is the focus, and the fixed line is the directrix of the parabola. As shown in Figure 6, the axis of symmetry of a parabola passes through the focus and is perpendicular to the directrix. The vertex is the midpoint of the line segment joining the focus and directrix on the axis. x y D(x, –p) p p P(x, y) Equal distance F(0, p) y = –p V(0, 0) d(P, F) = d(P, D) for all P on the parabola. Figure 7 x y Vertex Directrix Focus Axis of symmetry Figure 6 Notice that the focus is located “inside” the parabola. We can find an equation of a parabola from the preceding definition. Let p represent the directed distance from the vertex to the focus. Then the directrix is the line y = -p and the focus is the point F10, p2. See Figure 7. To find the equation of the set of points that are the same distance from the line y = -p and the point 10, p2, choose one such point P with coordinates 1x, y2. Because d1P, F2 and d1P, D2 must be equal, using the distance formula gives the following. d1P, F2 = d1P, D2 21x - 022 + 1y - p22 = 21x - x22 + 1y - 1-p222 Distance formula 2 x2 + 1y - p22 = 21y + p22 Simplify. x2 + y2 - 2yp + p2 = y2 + 2yp + p2 Square each side, and multiply. x2 = 4py Simplify. From this result, if the given form of the equation is y = ax2, then a = 1 4p . Remember the middle terms. Parabola with Vertical Axis of Symmetry and Vertex (0, 0) The parabola with focus 10, p2 and directrix y = -p has the following equation. x2 =4py This parabola has vertical axis of symmetry x = 0 and opens up if p 70 or down if p 60. x y y = –p 0 F(0, p) x 2 = 4py, p < 0 Axis x = 0 x y y = –p 0 Axis x = 0 F(0, p) x2 = 4py, p > 0
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