[PDF] 103 Ellipses

103 Ellipses

Section 10 3 Ellipses 745 When discussing ellipses, you might also choose to discuss the latera recta as background for Exercises 62–66 Consider the equation of the ellipse If you let then the equa-tion can be rewritten as which is the standard form of the equation of a circle with radius (see Section 1 2) Geometrically, when for

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Introduction

The second type of conic is called an ellipse,and is defined as follows. is constant.

FIGURE10.18FIGURE10.19

The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major axis,and its midpoint is the center of the ellipse. The chord perpendicular to the major axis at the center is the minor axisof the ellipse. See Figure 10.19. You can visualize the definition of an ellipse by imagining two thumbtacks placed at the foci, as shown in Figure 10.20. If the ends of a fixed length of string are fastened to the thumbtacks and the string is drawn tautwith a pencil, the path traced by the pencil will be an ellipse.

FIGURE10.20

To derive the standard form of the equation of an ellipse, consider the ellipse in Figure 10.21 with the following points: center, vertices, foci, Note that the center is the midpoint of the segment joining the foci. ?h±c, k?. ?h±a, k?;?h, k?; d 1 ?d 2

Major axis

MinoraxisCenter

Vertex

Vertex

FocusFocus(x,y)

d d 21

744Chapter 10 Topics in Analytic Geometry

Whatyou should learn

•Write equations of ellipses in standard form and graph ellipses. •Use properties of ellipses to model and solve real-life problems. •Find eccentricities of ellipses.

Whyyou should learn it

Ellipses can be used to model

and solve many types of real-life problems.For instance, in Exercise 59 on page 751,an ellipse is used to model the orbit of Halley's comet.

Ellipses

Harvard College Observatory/

SPL/Photo Researchers, Inc.

10.3

Definition of Ellipse

An ellipseis the set of all points in a plane, the sum of whose distances from two distinct fixed points (foci)is constant. See Figure 10.18.?x, y? b a c (,)h k(, ) x y bc 22
bc 22
2 b 2 + c 2 = 2a b 2 + c 2 = a 2

FIGURE10.21

333202_1003.qxd 12/8/05 9:01 AM Page 744

The sum of the distances from any point on the ellipse to the two foci is constant.

Using a vertex point, this constant sum is

Length of major axis

or simply the length of the major axis. Now, if you let be anypoint on the ellipse, the sum of the distances between and the two foci must also be

That is,

Finally, in Figure 10.21, you can see that which implies that the equation of the ellipse is You would obtain a similar equation in the derivation by starting with a vertical major axis. Both results are summarized as follows. Figure 10.22 shows both the horizontal and vertical orientations for an ellipse.

Major axis is horizontal. Major axis is vertical.

FIGURE10.22

x (x-h) 2 (y-k) 2 2 b 2 a+= 1 2a

2b(h,k)

y (x-h) 2 (y-k) 2 2 a 2 b+= 1

2a2b(h,k)

y x ?x?h? 2 a 2 ?y?k? 2 b 2 ?1. b 2 ?x?h? 2 ?a 2 ?y?k? 2 ?a 2 b 2 b 2 ?a 2 ?c 2 ??x??h?c?? 2 ??y?k? 2 ???x??h?c?? 2 ??y?k? 2 ?2a.2a. ?x, y?? x, y?? a?c???a?c??2a

Section 10.3 Ellipses745

When discussing ellipses,you might also

choose to discuss the latera recta as background for Exercises 62-66.

Consider the equation of the

ellipse

If you let then the equa-

tion can be rewritten as which is the standard form of the equation of a circle with radius (see Section 1.2).

Geometrically, when for

an ellipse, the major and minor axes are of equal length, and so the graph is a circle.a?br?a ?x?h? 2 ??y?k? 2 ?a 2 a?b, ?x?h? 2 a 2 ?y?k? 2 b 2 ?1.

Standard Equation of an Ellipse

The standard form of the equation of an ellipse,with center and major and minor axes of lengths and respectively, where is

Major axis is horizontal.

Major axis is vertical.

The foci lie on the major axis, units from the center, with If the center is at the origin the equation takes one of the following forms. x 2 b 2 ?y 2 a 2 ?1x 2 a 2 ?y 2 b 2 ?1 ?0, 0?,c 2 ?a 2 ?b 2 .c ?x?h? 2 b 2quotesdbs_dbs20.pdfusesText_26