682 CHAPTER 10 Analytic Geometry OBJECTIVES 1 Analyze Parabolas with Vertex at the Origin (p. 682) 2 Analyze Parabolas with Vertex at h k , ( ) (p. 686) 3 Solve Applied Problems Involving Parabolas (p. 688) 10.2 The Parabola Now Work the ‘Are You Prepared?’ problems on page 689. • Distance Formula (Section 1.2, p. 14) • Symmetry (Section 1.3, pp. 21–23) • Square Root Method (Section A.6, p. A49) • Completing the Square (Section A.3, p. A29) • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) • Quadratic Functions and Their Properties (Section 3.3, pp. 157–166) PREPARING FOR THIS SECTION Before getting started, review the following: Figure 3 Parabola a a 2a Axis of symmetry F P d(F, P) d(P, D) V Directrix D In Section 3.3, we learned that the graph of a quadratic function is a parabola. In this section, we give a geometric definition of a parabola and use it to obtain an equation. DEFINITION Parabola A parabola is the collection of all points P in a plane that are the same distance d from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix . As a result, a parabola is the set of points P for which ( ) ( ) = d F P d P D , , (1) Figure 3 shows a parabola (in blue). The line through the focus F and perpendicular to the directrix D is the axis of symmetry of the parabola.The point of intersection of the parabola with its axis of symmetry is the vertex V. Because the vertex V lies on the parabola, it must satisfy equation (1): ( ) ( ) = d F V d V D , , . The vertex is midway between the focus and the directrix. We let a equal the distance ( ) d F V, from F to V. To derive an equation for a parabola, we use a rectangular system of coordinates positioned so that the vertex V, focus F, and directrix D of the parabola are conveniently located. 1 Analyze Parabolas with Vertex at the Origin If we locate the vertex V at the origin ( ) 0, 0 , we can conveniently position the focus F on either the x -axis or the y -axis. First, consider the case where the focus F is on the positive x -axis, as shown in Figure 4. Because the distance from F to V is a, the coordinates of F will be ( ) a, 0 with > a 0. Similarly, because the distance from V to the directrix D is also a, and because D is perpendicular to the x -axis (since the x -axis is the axis of symmetry), the equation of the directrix D is = − x a. Now, if ( ) = P x y , is any point on the parabola, then P satisfies equation (1): ( ) ( ) = d F P d P D , , So we have ( ) ( ) ( ) ( ) − + − = + − + = + − + + = + + = x a y x a x a y x a x ax a y x ax a y ax 0 2 2 4 2 2 2 2 2 2 2 2 2 2 2 Use the Distance Formula. Square both sides. Multiply out. Simplify. Figure 4 y (0, 0) P 5 (x, y) V d(F, P) d(P, D) F 5 (a, 0) (2a, y) D: x 5 2a x
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