Chapter 2: Introduction to Implicit Equations

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1 Habeman MTH 11 Section V: Paametic and Implicit Equations Chapte : Intoduction to Implicit Equations When we descibe cuves on the coodinate plane with algebaic equations, we can define the elationship between the x- and y-coodinates of the cuve eithe explicitly o implicitly To explain the diffeence, let s conside the equation y x 7 The equation y x 7 establishes a elationship between x and y We say that y is defined explicitly in tems of x since the equation gives us a vey clea desciption of the elationship between x and y: no matte what the x-value is, the coesponding y- value is always x 7 If we subtact x fom each side of this equation we obtain the equation x y 7 Clealy this equation descibes the same elationship between x and y as the equation y x 7, but when y isn t isolated, the elationship between the vaiables isn t explicit Instead, this elationship is implied by the equation, so we say that it is an implicit equation An equation involving two vaiables is consideed explicit if one of the vaiables is isolated on one side of the equation, while and equation is consideed implicit if neithe of the vaiables is isolated on one side of the equation As we ve seen, equations like y x 7 can be witten eithe explicitly o implicitly, but many equations can only be witten implicitly Fo example, it s not possible to solve the implicit equation x y 1 fo eithe vaiable If we ty to solve the equation fo y, we need to use two explicit equations to communicate the infomation in the implicit equation: x y 1 y 1 x y 1 x y 1 x o y 1 x

2 EXAMPLE 1: Recall fom Example 4 in the pevious chapte that the system of paametic equations below descibe a unit cicle centeed at the point (0 :, 0) x cos( y sin( As we leaned in the pevious chapte, we use the Pythagoean Theoem to eliminate the paamete t and tansfom this system into a single implicit equation: cos ( sin ( 1 x y 1 Thus, the implicit equation x y 1 descibes a unit cicle centeed at the point (0, 0) Since the paametic system x cos( y sin( descibes a cicle of adius 1 unit, we should expect that if we multiply both the x- and y- coodinate by the facto we will stetch (o compess) the cicle so that its adius will change fom 1 unit to units, and if we add constants h and k to the x- and y-coodinates, espectively, we will shift the cente of the cicle fom the point (0, 0) to the point ( h, k) A genealized paameteization of a cicle is given in the box below If h, k, and 0 then the system of paametic equations below defines a cicle of adius centeed at the point ( h, k) x h cos( y k sin(

3 EXAMPLE : Eliminate the paamete t fom the system of paametic equations given in the box above to obtain an implicit equation that defines a cicle of adius centeed at the point ( h, k) We can use the Pythagoean Theoem to eliminate the paamete just as we did in Example 1 The Pythagoean Theoem involves sin( and cos( so we need to fist we need to isolate sin( and co s( in the equations in ou system: x h cos( cos( x h x h cos( and y k sin( sin( y k y k sin( Now, we can substitute the expessions x h and y k fo cos( t ) and sin( in the Pythagoean Theoem and obtain an implicit equation fo the cicle: cos ( sin ( 1 x h y k x h y k x h y k 1 1 If h, k, and 0 then the implicit equation x h y k defines a cicle of adius centeed at the point ( h, k)

4 4 EXAMPLE : Detemine the cente and adius of the cicle defined by the implicit equation ( x 7) ( y 6) 9 Notice that the equation ( x 7) ( y 6) 9 has the fom x h y k whee h 7, k 6, and Thus, the cente of the cicle is at the point ( 7, 6) and the adius is units We could check if we ae coect by gaphing the cicle on ou gaphing calculato, but ou calculatos aen t able to gaph implicit equations If we tansfom ou implicit equation into a system of paametic equations, we could use ou gaphing calculato to gaph the cicle To tansfom the implicit equation into a system of paametic equations, we mimic what we did in Example but do eveything in the opposite ode Fist, let s get 1 on the ight side of the equation by dividing both sides by (ie, 9): ( x 7) ( y 6) 9 ( x 7) ( y 6) ( x 7) ( y 6) 1 x 7 y 6 1 a We now have an equation in which the sum of two squaes is equal to 1 Since Pythagoean Theoem has this same fom (ie, since cos ( sin ( t ) 1), we can let x 7 y 6 cos( and sin( (Notice that if we substitute these values fo cos( and sin( in the Pythagoean Theoem we obtain the equation labeled a above) We can solve these equations fo x and y to obtain a paameteization of the given implicit equation: x 7 cos( x 7 cos( x 7 cos( and y 6 sin( y 6 sin( y 6 sin( Thus, the system of paametic equations below descibes the same cicle as the implicit equation ( x 7) ( y 6) 9 x 7 cos( y 6 sin(

5 5 Notice that fom the genealization given at the beginning of Example, we can see that this paameteization descibes a cicle of adius units with cente ( 7, 6) which is exactly what we wee looking fo EXAMPLE 4: The gaph of the equation x y 6x 4y is a cicle Identify the cente and adius (by using appopiate algeba) In ode to detemine the cente and adius of the cicle, we need to get ou implicit equation into the fom x h y k To accomplish this, we can complete the squae twice (If you need to eview the pocedue fo completing the squae, see page A40 in the appendix of ou textbook as well as page 144 in in ou textbook) x y 6x 4y x 6x y 4y x 6x 9 9 y 4y 4 4 ( x 6x 9) ( y 4y 4) 9 4 ( x ) ( x ) 1 ( x ) ( x ) 16 ( x ) ( x ) 4 Thus, the cente of the cicle is at the point (, ) and the adius of the cicle is 4 units Based on what we obseved in Examples and 4, we can conclude that the system of paametic equations below descibes this same cicle x 4cos( y 4sin( Gaph the system on you gaphing calculato to check

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal