Quadratic Equations and Conics A quadratic equation in two variables is an equation that’s equivalent to an equation of the form p(x, y)=0 where p(x, y)isaquadraticpolynomial. Examples. 4x2 3xy 2y2 + x y +6=0isaquadraticequation,asare •x2 y−2 =0and− x2 + y−2 =0andx2 1=0. − − y = x2 is a quadratic equation. It’s equivalent to y x2 =0, •and y x2 is a quadratic polynomial. − − xy =1isaquadraticequation.It’sequivalenttothequadratic •equation xy 1=0. − x2 +y2 = 1isaquadraticequation.It’sequivalenttox2 +y2 +1 = 0. • − x2 + y = x2 +2isanotaquadraticequation.It’salinearequation. •It’s equivalent to y 2=0,andy 2isalinearpolynomial. − − A conic is a set of solutions of a quadratic equation in two variables. In contrast to lines–solutions of linear equations in two variables–it takes a fair amount of work to list all of the possible geometric shapes that can possibly arise as conics. In what remains of this chapter, we’ll take a tour of some conics that we already know. After we cover trigonometry in this course, we’ll return to conics and explain all of the possible shapes of conics. 154 Parabola The graph of the function in one variable f(x) = x2 is called a parabola. No points We saw in the chapter PolynomialJ Equations that the quadratic equation x2 + y2 = 1 − has no solution. A sum of squares can never be negative. Thus, is an example of conic. ; x2 = 1hasnosolutiontoo! − The poilits iii the graphOneof pointf(x) are points of the form (x, f(x)). That is, the poillts in the graph haveWe sawthe inform the chapter(x, x2). PolynomialThese are the Equationspoints in thatthe the quadratic equation plane whose second coordinates are the square of theirxfirst2 + ycoordinates,2 =0 or in other words, points of the form (x, y) where has exactly one solution, the point (0, 0). The only way to take two numbers, square them,y take= the sum of those squares, and get 0, is if the two numbers To recap, the graph youof f(x) start= withx2 in arethe 0.plane Thus,is the thesame set containingset as the theset single point (0, 0) is a of solutions of the quadraticconic.equation in two variables given by the equation = x2 0 I 150 155 ParallelParallellines lines AnAnequation equationfor forthe thevertical verticalline linethat thatcrosses crossesthe thex-axisx-axisat atthe thenumber number1 1 isis x==1,orequivalently,1, equivalently, xx— 11 = 0.Similarly, Similarly,xx +1=0isanequationfor1 = 0 is an equation for x or − 0. + thethevertical verticalline linethat thatcrosses crossesthe thex-axisx-axisat atthe thenumber number—1.1.As Aswe wesaw sawin inthe the − previouspreviouschapter, chapter,the theunion unionof ofthese thesetwo twovertical verticallines linesis isthe theset setof ofsolutions solutions ofofthe theequation equation (x— 1)(x1)(x +1)=01) = (x − + 0 ananequation equationthat thatcan canbe bewritten writtenmore moresimply simplyas as x21—Ox2 1=0 − Thus,Thus,two twoparallel parallellines linescami canappear appearas asa aconic. conic. —1 Parallel lines IntersectingIntersectingAn equation forlines linesthe vertical line that crosses t.he x-axis at the number 1 = — = isIt’sIt’sx also also1, possibleor possibleequivalently,to tohave havextwo two1intersecting intersecting0. Similarly,lines linesxas as+a a1 conic. conic.0 is In Inanthe theequationprevious previousfor chapterchapterthe verticalwe wesaw sawhuethat thatthatthe thecrossessolutions solutionsthe x-axisof of at the number —1. As we saw in the previous chapter, the union of these two vertical lines is the set of solutions x2— y2 ==0 of the equation − 0 are a union of the two lines that have slope 1 and 1, and that each pass are a union of the two lines that(x — have1)(x +slope1) =1 0and —1, and that each pass through the point (0, 0). Recall that x y =0isanequationforthefirstof− throughan equationthe pointthat can(0, 0).be Recallwrittenthatmorex —simplyy = 0 asis an equation for the first of these two lines, and that x + y==0isanequationforthesecondofthesetwo− these two lines, and that x + y 0 is an equation= for the second of these two lines. Therefore, (x y)(x + y)= = 0 is—1 an equation0 for the union of these two lines. Therefore, (x —−y) (x + y) 0 is an equation for the union of these two lines,Thus, andtwo itparallel can belines simplifiedcan appear as x2 asya2 ==0.conic. lines, and it can be simplified as —− 0. / N 156 Intersecting lines 149 It’s also possible to have two intersecting lines as a conic. In the previous chapter we saw that the solutions of 2 —y_=O are a union of the two lines that have slope 1 and —1, and that each pass through the point (0, 0). Recall that x — y 0 is an equation for the first of these two hues, and that x + y = 0 is an equation for the second of these two lines. Therefore, (x — y)(x + y) = 0 is an equation for the union of these two lines, and it can be simplified as x2 — y2 = 0. 149 Single line x2 =0isaquadraticequation,soitssetofsolutionsintheplaneisaconic. The only number whose square is 0 is 0 itself, so if x2 =0,thenx =0. Thus, the set of solutions of x2 = 0 are those points in the plane whose x- coordinates equal 0. This is a single vertical line, the y-axis. This single line is an example of a conic. _____ __ ___ Parabola The graph ofParabolathe function in one variable f(x) = x2 is called a parabola. The graph of the function in one variable f(x)=x2 is called a parabola. J(x=x2 The points in the graph of f(x)arepointsoftheform(x, f(x)). That is, The points inthethe pointsgraph inof thef(x) graphare points have theof the formform (x,(x, x2).f(x)). TheseThat areis, the points in the the points in theplanegraph whosehave secondthe form coordinates(x, x2). areThese theare squarethe ofpoints theirin firstthe coordinates, or plane whose secondin othercoordinates words, pointsare the of thesquare formof (x,their y)wherefirst coordinates, or in other words, points of the form (x, y) where y = x2 = 157 = To recap, the graph of f(x) x2 in the plane is the same set as the set of solutions of the quadratic equation in two variables given by the equation = J 150 _______ _______ _______ _______ _______ _______ Parabola The graph of the function in one variable f(x) = x2 is called a parabola. To recap, the graph of f(x)=x2 in the plane is the same set as the set Parabola of solutions of the quadratic equation in two variables given by the equation 2 The graph ofythe= xfunction. That is,in theone parabolavariable f(x) below= isx2 ais conic.called a parabola. The points in the graph of f(x) are points of the form (x, f(x)). That is, the points in the graph have the form (x, x2). These are the points in the plane whose second coordinates are the square of their first coordinates, or in other words, points of the form (x, y) Iiere The points in the graph of f(x) are points of the form (x, f(x)). That is,= the points in the graph have the form x2). These are Hyperbola (x, the points in= the plane whose second coordinates are the squareTo recap,of theirthe graphfirst of f(x) x2orin the plane is the same set as the set coordinates,1 1 in other words, pointsThe pointsof the inform the graph(x, y)ofIiere ofsolutions the functionof thegquadratic(x)=x areequation of the formin two (x,variablesx). given by the equation = y=12. = To recap, the graph of f(x) x2 in the plane is the same set as the set .of solutions of the quadratic equation in two variables given by the equation y=12 JA(2)(s) JA(2)(s) a a That is, they are points whose y-coordinates are the multiplicative150 inverse of 1 their x-coordinates. In other words, the graph of g(x)=x is exactly the set 1 of points (x, y)intheplanewherey = x. 158 150 _____ __ ___ Parabola The graph of the function in one variable f(x) = x2 is called a parabola. J(x=x2 The points in the graph of f(x) are points of the form (x, f(x)). That is, the points in the graph have the form (x, x2). These are the points in the 1 We know from the domainplane ofwhose the functionsecond coordinatesg(x)=x thatare xthe=0.Thus,wesquare of their first coordinates, or 1 6 can multiply the equationin othery = xwords,by x topoints obtainof thethe equivalentform (x, y) equationwhere xy =1 = 1 = x2 so that we now have thatTo therecap, graphthe of xgraphis exactlyof f(x) the set ofin solutionsthe plane of theis the same set as the set equation xy =1. of solutions of the quadratic equation in two variables given by the equation = J The equation xy =1isaquadraticequation,soitssetofsolutionsisa150 conic. It’s called a hyperbola. 159 _______ _______ _______ Exercises ParabolaParabola #1-5 involve writing equations for planar transformations of the parabola that is the set of solutions of y = x2. We’ll call this conic S. The graph of the function in one variable f(x) = x2 is called The graph of the function in one variable f(x) = x2 is calledJ2a parabola.a parabola. The points in the graph of are points of the form The points in the graph1.)ofThef(x) parabolaaref()points shiftedof the form right(x, 2f(x)). and(x, upThatf(x)). 4: is, That is, the points in the graph have x2). 2 2 the form (x, These are the points in the the points in the graphRecallhave thatthe form the planar(x, x2).