Analytic Geometry in Two and Three Dimensions

1 631 Analytic Geometry in Twoand Three Sections andRotation of Equations SystemCHAPTER8 The oval-shaped lawn behind the White House inWashington, is called the Ellipse. It has views of theWashington Monument, the Jefferson Memorial, theDepartment of Commerce, and the Old Post Office Ellipse is 616 ft long, 528 ft wide, and is in the shape ofa conic section. Its shape can be modeled using the methodsof this chapter. See page 1/13/06 6:49 AM Page 631 Chapter 8 OverviewAnalytic Geometry combines number and form . It is the marriage of algebra and geom-etry that grew from the works of Frenchmen Ren Descartes (1596 1650) and Pierrede Fermat (1601 1665).

2 Their achievements allowed Geometry problems to be solvedalgebraically and algebra problems to be solved geometrically two major themes ofthis book. Analytic Geometry opened the door for Newton and Leibniz to develop Sections , we will learn that parabolas, ellipses, and hyperbolas are all conicsections and can all be expressed as second-degree equations. We will investigate theiruses, including the reflective properties of parabolas and ellipses and how hyperbolasare used in long-range navigation. In Section , we will see how parabolas, ellipses,and hyperbolas are unified in the polar-coordinate setting.

3 In Section , we will movefrom the two-dimensional plane to revisit the concepts of point, line, midpoint, dis-tance, and vector in Three -dimensional 8 Analytic Geometry in Two and Three Sections and ParabolasWhat you ll learn about Conic Sections Geometry of a Parabola Translations of Parabolas Reflective Property of aParabola.. and whyConic sections are the paths ofnature: Any free-moving objectin a gravitational field followsthe path of a conic SectionsImagine two nonperpendicular lines intersecting at a point V. If we fix one of thelines as an axisand rotate the other line (the generator) around the axis, then thegenerator sweeps out a with V, as illustrated in Figure that Vdivides the cone into two parts called , with each nappe of thecone resembling a pointed ice-cream A right circular cone (of two nappes).

4 A (or ) is a cross section of a cone, in other words, the intersectionof a plane with a right circular cone. The Three basic conic sections are the parabola, theellipse, and the hyperbola(Figure ).Some atypical conics, known as , are shown in Figure it is atypical and lacks some of the features usually associated with an ellipse,degenerate conic sectionsconicconic sectionAxisGeneratorUppernappeLowernappe Vnappesvertexright circular coneHISTORY OF CONIC SECTIONSP arabolas, ellipses, and hyperbolas hadbeen studied for many years whenApollonius (c. 262 190 ) wrote hiseight-volume Conic , born in northwestern AsiaMinor, was the first to unify thesethree curves as cross sections of a coneand to view the hyperbola as havingtwo branches.

5 Interest in conic sec-tions was renewed in the 17th centurywhen Galileo proved that projectilesfollow parabolic paths and JohannesKepler (1571 1630) discovered thatplanets travel in elliptical 1/13/06 6:49 AM Page 632a circle is considered to be a degenerate ellipse. Other degenerate conic sections can beobtained from cross sections of a degenerate cone; such cones occur when the genera-tor and axis of the cone are parallel or perpendicular. (See Exercise 73.)SECTION Sections and Parabolas633 Ellipse(a)(b)ParabolaHyberbolaPoint: plane throughcone's vertex onlySingle line: planetangent to coneIntersecting linesFIGURE (a) The Three standard types of conic sections and (b) Three degenerate conic sections can be defined algebraically as the graphs of , that is, equations of the formAx2 Bxy Cy2 Dx Ey F 0,where A,B, and Care not all zero.

6 ( quadratic ) equations in two variablessecond-degreeOBJECTIVES tudents will be able to find the equation,focus, and directrix of a students if all parabolas are similar(in the geometric sense). (Yes)LESSON GUIDEDay 1: Conic Sections; Geometry of aParabolaDay 2: Translations of Parabolas;Reflective Property of a ParabolaBIBLIOGRAPHYFor students:Practical Conic Sections,J. W. Downs. Dale Seymour Publications, , Carl Sagan and Ann Books, teachers: Astronomy: From the Earthto the Universe (5th ed.),J. M. Pasachoff. Saunders CollegePublishing, and Gender Equity in theMathematics Classroom: The Gift ofDiversity (1997 Yearbook),JanetTrentacosta (Ed.)

7 National Council ofTeachers of Mathematics, :Conic SeymourPublicationsLocating Satellites in Elliptical Orbits,National Council of Teachers ofMathematicsTEACHING NOTESYou may wish to construct the nappe of acone by rolling up a piece of paper inorder to illustrate various conic sectionsas cross sections of a that a circle is a special case of anellipse and thus considered a 1/13/06 6:49 AM Page 633 Geometry of a ParabolaIn Section we learned that the graph of a quadratic function is an upward or down-ward opening parabola. We have seen the role of the parabola in free-fall and projec-tile motion.

8 We now investigate the geometric properties of 8 Analytic Geometry in Two and Three DimensionsDEFINITIONP arabolaA is the set of all points in a plane equidistant from a particular line (the) and a particular point (the ) in the plane. (See Figure )focusdirectrixparabolaThe line passing through the focus and perpendicular to the directrix is the (focal)of the parabola. The axis is the line of symmetry for the parabola. The point where theparabola intersects its axis is the of the parabola. The vertex is located midwaybetween the focus and the directrix and is the point of the parabola that is closest toboth the focus and the directrix.

9 See Figure vertexaxisFIGURE of a Parabola. The distance from each point on the parabola to boththe focus and the directrix is the on the parabolaDist. to focusAxisFocusVertexDirectrixDist. to directrixEXPLORATION 1 Understanding the Definition of that the vertex of the parabola with focus 0, 1 and directrixy 1 is 0, 0 . (See Figure ) an equation for the parabola shown in Figure the coordinates of the points of the parabola that are highlighted inFigure DEGENERATE PARABOLAIf the focus Flies on the directrix l, theparabola degenerates to the linethrough Fperpendicular to , we will assume Fdoes notlie on OF A POINTB efore the word setwas used in math-ematics, the Latin word locus, meaning place, was often used in geometricdefinitions.

10 The locus of a point wasthe set of possible places a point couldbe and still fit the conditions of thedefinition. Sometimes, conics are stilldefined in terms of NOTEE xercise 71 explains how Figure wascreated, and this exercise can be usedinstead of (or in addition to) Exploration EXTENSIONSFind the y-coordinate for a point on theparabola that is a distance of 25 unitsfrom the focus. Ans. 24yx(0, 1)FIGURE Geometry of a 1/13/06 6:49 AM Page 634We can generalize the situation in Exploration 1 to show that an equation for the parab-ola with focus 0,p and directrix y pis x2 4py.