B.1 Conic Sections - Cengage

1 12/27/11 2:01 PM Page B1. Appendix Conic Sections B1. B Conic Sections Conic Sections Recognize the four basic conics: circles, parabolas, ellipses, and hyperbolas. Recognize, graph, and write equations of parabolas (vertex at origin). Recognize, graph, and write equations of ellipses (center at origin). Recognize, graph, and write equations of hyperbolas (center at origin). Introduction to Conic Sections Conic Sections were discovered during the classical Greek period, which lasted from 600 to 300 By the beginning of the Alexandrian period, enough was known of conics for Apollonius (262 190 ) to produce an eight-volume work on the subject.

2 This early Greek study was largely concerned with the geometric properties of conics. It was not until the early seventeenth century that the broad applicability of conics became apparent. A Conic section (or simply Conic ) can be described as the intersection of a plane and a double-napped cone. Notice from Figure that in the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone. When the plane does pass through the vertex, the resulting figure is a degenerate Conic , as shown in Figure Conic Sections Degenerate Conics FIGURE FIGURE There are several ways to approach the study of conics.

3 You could begin by defining conics in terms of the intersections of planes and cones, as the Greeks did, or you could define them algebraically, in terms of the general second-degree equation Ax 2 Bxy Cy 2 Dx Ey F 0. However, you will study a third approach in which each of the conics is defined as a locus, or collection, of points satisfying a certain geometric property. For example, in section you saw how the definition of a circle as the collection of all points x, y that are equidistant from a fixed point h, k led easily to the standard equation of a circle, x h 2 y k 2 r 2.

4 You will restrict your study of conics in Appendix to parabolas with vertices at the origin, and ellipses and hyperbolas with centers at the origin. In Appendix , you will look at the general cases. 12/27/11 2:01 PM Page B2. B2 Appendix B Conic Sections Parabolas In section , you determined that the graph of the quadratic function given by STUDY TIP f x ax2 bx c Note that the term parabola is a parabola that opens upward or downward. The definition of a parabola given below is a technical term used in is more general in the sense that it is independent of the orientation of the parabola.

5 Mathematics and does not simply refer to any U-shaped curve. Definition of a Parabola y A parabola is the set of all points x, y in a plane that are equidistant from a fixed line called the directrix and a fixed point called the focus d2. (not on the line). The midpoint Focus (x, y). between the focus and the directrix d1. x is called the vertex, and the line Vertex d1 d2. passing through the focus and the vertex is called the axis of the parabola. Directrix Axis Using this definition, you can derive the following standard form of the equation of a parabola. Standard Equation of a Parabola (Vertex at Origin).

6 The standard form of the equation of a parabola with vertex at 0, 0 and directrix y p is given by x2 4py, p 0. Vertical axis For directrix x p, the equation is given by y2 4px, p 0. Horizontal axis The focus is on the axis p units (directed distance) from the vertex. See Figure y y x 2 = 4py, p 0. Vertex (0, 0) (x, y). Focus (0, p). Vertex p (0, 0). (x, y) Focus (p, 0). x x Axis Directrix: x = p p p p y 2 = 4px, p 0. Directrix: y = p Axis Parabola with Vertical Axis Parabola with Horizontal Axis FIGURE 12/27/11 2:01 PM Page B3. Appendix Conic Sections B3. Example 1 Finding the Focus of a Parabola Find the focus of the parabola whose equation is y 2x2.

7 SOLUTION Because the squared term in the equation involves x, you know that the axis is vertical, and the equation is of the form y x2 4py. Standard form, vertical axis (. Focus 0, 18 (. x You can write the original equation in this form, as shown. 1 1. 2x2 y Write original equation. y = 2x2. 1. 1 x2 y Divide each side by 2. 2. 2 81 y. x2 4 Write in standard form. So, p 18. Because p is negative, the parabola opens downward and the focus of the FIGURE parabola is 0, p 0, 18 , as shown in Figure Checkpoint 1. Find the focus of the parabola whose equation is y2 2x.))

8 Y Example 2 Finding the Standard Equation of a Parabola 2. y2 = 8x Write the standard form of the equation of the parabola with vertex at the origin and 1 focus at 2, 0 . Focus Vertex (2, 0) SOLUTION The axis of the parabola is horizontal, passing through 0, 0 and 2, 0 , as x 1 2 3 4. shown in Figure So, the standard form is (0, 0). 1 y2 4px. Standard form, horizontal axis Because the focus is p 2 units from the vertex, the equation is 2. y2 4 2 x Standard form FIGURE y2 8x. Checkpoint 2. Write the standard form of the equation of the parabola with vertex at the origin and focus at 0, 2.

9 Parabolas occur in a wide variety of Light source applications. For instance, a parabolic at focus reflector can be formed by revolving a parabola about its axis. The resulting Focus surface has the property that all incoming Axis rays parallel to the axis are reflected through the focus of the parabola. This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversely, the light rays emanating from the focus of the parabolic reflector used Parabolic reflector: in a flashlight are all reflected parallel to Light is reflected in parallel rays.

10 One another, as shown in Figure FIGURE 12/27/11 2:01 PM Page B4. B4 Appendix B Conic Sections Ellipses Another basic type of Conic is called an ellipse. Definition of an Ellipse An ellipse is the set of all points x, y in a plane the sum of whose distances from two distinct fixed points, called foci, is constant. (x, y). d1 d2. Major axis Center Vertex Vertex Focus Focus Minor axis d 1 + d 2 is constant. The line through the foci intersects the ellipse at two points, called the vertices. The chord joining the vertices is called the major axis, and its midpoint is called the center of the ellipse.