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
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155 ParallelParallellines lines AnAnequation equationfor forthe thevertical verticalline linethat thatcrosses crossesthe thex-axisx-axisat atthe thenumber number1 1 isisxx ==1,orequivalently,1, or equivalently, xx — 11 =0. 0.Similarly, Similarly,xx +1=0isanequationfor1 0 is an equation for + = thethevertical verticalline line thatcrosses crossesthe thex-axisx-axisat atthe thenumber number—1.1.As Aswe wesaw sawin inthe the that previouspreviouschapter, chapter,the theunion unionof ofthese thesetwo twovertical verticallines linesis isthe theset setof ofsolutions solutions ofofthe theequation equation (x(x— 1)(x1)(x +1)=01) 0 + = ananequation equationthat thatcan canbe bewritten writtenmore moresimply simplyas as x2 1=0 1—O Thus,Thus,two twoparallel parallellines linescami canappear appearx2 as asa aconic. conic.
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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 two— 1intersecting intersecting0. Similarly,lines linesxas as+ a a1conic. conic.= 0 isIn 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 arearea aunion unionof ofthe thetwo twolines linesthat that(x — have have1)(x +slope slope1) = 1 10and and —1,1,and andthat thateach eachpass pass through the point (0, 0). Recall that x y =0isanequationforthefirstof throughan equationthe pointthat can(0, 0).be Recallwrittenthatmorex — simply y = 0asis an equation for the first of thesethesetwo twolines, lines,and andthat thatxx++yy ==0isanequationforthesecondofthesetwo0 is an equation for the second of these two —1 = 0 lines. Therefore, (x y)(x + y) =0 0is is an equationfor for the union of these two lines. Therefore, (x — y) (x + y) = an equation 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.
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156 / 149 Intersecting lines 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.
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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 =
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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 form Hyperbola the (x, x2). These are the points in the plane whose second coordinates are the squareTo recap,of theirthe graphfirst coordinates,of f(x) = x2orin the plane is the same set as the set The points in the graph of the function g(x)= 1 are of the form (x, 1 ). in other words, points of the form (x, y)ofIieresolutions of the quadraticx equation in two variablesx given by the equation = 2y=1 = 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 .2y=1 J2A()(s) a J2A()(s) 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, These are the points in the 1 x2). 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 = so that we now have thatTo therecap, graphthe of xgraphis exactlyof f(x) the setx2 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 =
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The equation xy =1isaquadraticequation,soitssetofsolutionsisa150 conic. It’s called a hyperbola.
159 ______Exercises Parabola #1-5 involve writing equations for planar transformations of the parabola Parabola 2 The ofthat isfunction the setin of solutions of y = x . We’ll call this conic S. The graph ofgraphthe functionthe in one variableone variablef(x) = x2f(x)is called= x2 isa calledparabola.a parabola. J2
The points in the graph of are points of form The points in the graph1.)ofThef(x) parabolaaref()points shiftedof the form rightthe(x, 2f(x)). and(x, upThatf(x)). 4: is,That is, the points in the graph have the form (x, These are the 2points2 in the the points in the graphRecallhave thatthe form the planar(x, transformationThesex2). are theApoints(2,4) : Rin theR moves points in the plane whose second coordinates arex2). the sqjiare of their first coordinates,! or plane whose second coordinatesplane rightare 2 andthe upsquare 4. Thatof their is, A(2first,4)(x,coordinates, y)=(x+2,yor+4). Find an equation in other words, points of the form (x, y) xhere 2 in other words, points forof theA(2,form4)(S).(x, Toy) doIiere this, precompose the equation for S, the equation y = x , 1 = by A . The= inverse of A(2,4) is A( 2, 4) and A( 2, 4)(x, y)=(x 2,y 4). (2,4) To recap. the(Don’tgraph simplifyof f(x) your= answer.)in the plane is the same set as the set To recap, the graph of f(x) = x2 in thex2 plane is the same set as the set of solutionsof solutionsof the quadraticof the quadraticequationequationin two variablesin two variablesgiven bygiventhe equationby the equation .2y=1 y = x2. J2A()(s) a
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150 150 Parabola The graph of the function in one variable f(x) = x2 is called a parabola. J2
The points in the graph of f() 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 sqjiare of their first coordinates, or in other words, points of2.)theTheform parabola(x, y) xhere shifted left 3 and down 2:
= 2 2 The planar transformation A( 3, 2) : R R moves points150 in the plane left ! To recap. the graph 3anddown2.Thatis,of f(x) = x2 in the planeA(is 3the, 2)(samex, y)=(set xas the3,yset 2). Find an equation 1 of solutions of the quadraticfor A(equation3, 2)(S).in Totwo dovariables this, precomposegiven by thethe equationequation for S by A . The ( 3, 2) x2. y = inverse of A( 3, 2) is A(3,2). (Don’t simplify your answer.)