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). 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.

2 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. 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).

3 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. 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.

4 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.(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.

5 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.) 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. 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.

6 (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. 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.

7 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. (See Figure )SECTION Sections and Parabolas635yxx2 = 4pyDirectrix: y = pFocus(a)F(0, p)D(x, p)lP(x, y)ppThe vertex lieshalfway betweendirectrix and = 4pyDirectrix: y = pVertex at origin(b)F(0, p)FocusWe must show first that a point P(x,y) that is equidistant from F(0,p) and the liney psatisfies the equation x2 4py, and then that a point satisfying the equationx2 4pyis equidistant from F(0,p) and the line y p:Let P(x,y) be equidistant from F(0,p) and the line y p.

8 Notice that x 0 2 y p 2 distance from P x,y to F 0,p , and x x 2 y p 2 distance from P x,y to y these distances and squaring yields: x 0 2 y p 2 x x 2 y p 2x2 y p 2 0 y p y2 2py p2 y2 2py 4pyCombine like reversing the above steps, we see that a solution x,y of x2 4pyis equidistantfrom F 0,p and the line y equation x2 4pyis the of the equation of an upward or down-ward opening parabola with vertex at the origin. If p 0, the parabola opens upward; ifp 0, it opens downward. An alternative algebraic form for such a parabola is y ax2,where a 1 4p . So the graph of x2 4pyis also the graph of the quadratic functionf x the equation of an upward or downward opening parabola is written as x2 4py,the value pis interpreted as the of the parabola the directeddistancefrom the vertex to the focus of the parabola. A line segment with endpoints on a parabolais a of the parabola. The value 4p is the of the parabola thelength of the chord through the focus and perpendicular to the widthchordfocal lengthstandard formFIGURE of x2 4pywith (a) p 0 and (b) p that the standard form of the equa-tion of an upward or downward openingparabola is not the same as the standardform of a quadratic GAMEA dditional features of a parabola aredefined in Exercises 74 76.

9 The focalwidth is the length of the latus 1/13/06 6:49 AM Page 635 Parabolas that open to the right or to the left are inverse relationsof upward or down-ward opening parabolas. So equations of parabolas with vertex 0, 0 that open to theright or to the left have the standard form y2 4px. If p 0, the parabola opens to theright, and if p 0, to the left. (See Figure )636 CHAPTER 8 Analytic Geometry in Two and Three DimensionsParabolas with Vertex (0, 0) Standard equationx2 4pyy2 4px OpensUpward or To the right ordownwardto the left Focus 0,p p,0 Directrixy px p Axisy-axisx-axis Focal lengthpp Focal width 4p 4p See Figures and 1 Finding the Focus, Directrix,and Focal WidthFind the focus, the directrix, and the focal width of the parabola y 1 2 both sides of the equation by 2 yields the standard formx2 2y. The coefficient of yis 4p 2, and p 1 2. So the focus is 0,p 0, 1 2 . Because p 1 2 1 2, the directrix is the line y 1 2.

10 Thefocal width is 4p 2 try Exercise 2 Finding an Equation of a ParabolaFind an equation in standard form for the parabola whose directrix is the line x 2and whose focus is the point 2, 0 .SOLUTIONB ecause the directrix is x 2 and the focus is 2, 0 , the focal lengthis p 2 and the parabola opens to the left. The equation of the parabola in standardform is y2 4px, or more specifically,y2 try Exercise of ParabolasWhen a parabola with the equation x2 4pyor y2 4pxis translated horizontally byhunits and vertically by kunits, the vertex of the parabola moves from 0, 0 to h,k .(See Figure ) Such a translation does not change the focal length, the focal width,or the direction the parabola opens. yxy2 = 4pxDirectrix:x = pVertex(a)FocusF(p, 0)Oyxy2 = 4pxDirectrix:x = pVertex(b)FocusF(p, 0)OFIGURE of y2 4pxwith (a) p 0and (b) p ON EXAMPLESWhen solving a problem such as Example2, students should be strongly encouragedto draw a sketch of the given may wish to draw a sketch for thestudents to model this 1/13/06 6:49 AM Page 636 EXAMPLE 3 Finding an Equation of a ParabolaFind the standard form of the equation for the parabola with vertex (3, 4) and focus(5, 4).