Recall from Geometry that a circle can be determined by fixing a point (called the center) and a positive number (called the radius) as follows.

Transcription

1 98 Hooked on Conics 7. Circles Recall from Geometr that a circle can be determined b fiing a point called the center) and a positive number called the radius) as follows. Definition 7.. A circle with center h, k) and radius r > 0 is the set of all points, ) in the plane whose distance to h, k) is r. r, ) h, k) From the picture, we see that a point, ) is on the circle if and onl if its distance to h, k) is r. We epress this relationship algebraicall using the Distance Formula, Equation., as r = h) + k) B squaring both sides of this equation, we get an equivalent equation since r > 0) which gives us the standard equation of a circle. Equation 7.. The Standard Equation of a Circle: The equation of a circle with center h, k) and radius r > 0 is h) + k) = r. Eample 7... Write the standard equation of the circle with center, ) and radius 5. Solution. Here, h, k) =, ) and r = 5, so we get )) + ) = 5) + ) + ) = 5 Eample 7... Graph + ) + ) =. Find the center and radius. Solution. From the standard form of a circle, Equation 7., we have that + is h, so h = and is k so k =. This tells us that our center is, ). Furthermore, r =, so r =. Thus we have a circle centered at, ) with a radius of. Graphing gives us

2 7. Circles 99 If we were to epand the equation in the previous eample and gather up like terms, instead of the easil recognizable + ) + ) =, we d be contending with = 0. If we re given such an equation, we can complete the square in each of the variables to see if it fits the form given in Equation 7. b following the steps given below. To Write the Equation of a Circle in Standard Form. Group the same variables together on one side of the equation and position the constant on the other side.. Complete the square on both variables as needed.. Divide both sides b the coefficient of the squares. For circles, the will be the same.) Eample 7... Complete the square to find the center and radius of = 0. Solution = = add to both sides ) + + ) = factor out leading coefficients + ) ) ) = + ) + complete the square in, 9 9 ) + + ) = 5 factor ) + + ) = 5 divide both sides b 9 From Equation 7., we identif as h, so h =, and + as k, so k =. Hence, the center is h, k) =, ). Furthermore, we see that r = 5 9 so the radius is r = 5.

3 500 Hooked on Conics It is possible to obtain equations like ) + + ) = 0 or ) + + ) =, neither of which describes a circle. Do ou see wh not?) The reader is encouraged to think about what, if an, points lie on the graphs of these two equations. The net eample uses the Midpoint Formula, Equation., in conjunction with the ideas presented so far in this section. Eample 7... Write the standard equation of the circle which has, ) and, ) as the endpoints of a diameter. Solution. We recall that a diameter of a circle is a line segment containing the center and two points on the circle. Plotting the given data ields r h, k) Since the given points are endpoints of a diameter, we know their midpoint h, k) is the center of the circle. Equation. gives us + h, k) =, ) + + =, + ) =, 7 ) The diameter of the circle is the distance between the given points, so we know that half of the distance is the radius. Thus, r = ) + ) Finall, since = )) + ) = + 0 = ) 0 = 0, our answer becomes + ) 7 ) = 0

4 7. Circles 50 We close this section with the most important circle in all of mathematics: the Unit Circle. Definition 7.. The Unit Circle is the circle centered at 0, 0) with a radius of. The standard equation of the Unit Circle is + =. Eample Find the points on the unit circle with -coordinate Solution. We replace with in the equation + = to get + + = ) = + =. Our final answers are, ) and, = = ± = ± ). While this ma seem like an opinion, it is indeed a fact. See Chapters 0 and for details.

5 50 Hooked on Conics 7.. Eercises In Eercises - 6, find the standard equation of the circle and then graph it.. Center, 5), radius 0. Center, ), radius. Center, 7 ), radius. Center 5, 9), radius ln8) 5. Center e, ), radius π 6. Center π, e ), radius 9 In Eercises 7 -, complete the square in order to put the equation into standard form. Identif the center and the radius or eplain wh the equation does not represent a circle = = = = = = In Eercises - 6, find the standard equation of the circle which satisfies the given criteria.. center, 5), passes through, ). center, 6), passes through, ) 5. endpoints of a diameter:, 6) and, ) 6. endpoints of a diameter:, ),, ) 7. The Giant Wheel at Cedar Point is a circle with diameter 8 feet which sits on an 8 foot tall platform making its overall height is 6 feet. Find an equation for the wheel assuming that its center lies on the -ais. 8. Verif that the following points lie on the Unit Circle: ±, 0), 0, ±), ) and ±, ± ) ±, ±, ±, ± 9. Discuss with our classmates how to obtain the standard equation of a circle, Equation 7., from the equation of the Unit Circle, + = using the transformations discussed in Section.7. Thus ever circle is just a few transformations awa from the Unit Circle.) 0. Find an equation for the function represented graphicall b the top half of the Unit Circle. Eplain how the transformations is Section.7 can be used to produce a function whose graph is either the top or bottom of an arbitrar circle.. Find a one-to-one function whose graph is half of a circle. Hint: Think piecewise.) ) Source: Cedar Point s webpage.

6 7. Circles Answers. + ) + + 5) = ) + + ) = ) + 7 ) =. 5) + + 9) = ln8)) 5 ln8) ln8) ln8) 9 9 ln8) 5. + e) + ) = π + π 6. π) + e ) = 9 e + 9 e π e e + π π e e 9 π 9 π π + 9

7 50 Hooked on Conics 7. ) + + 5) = Center, 5), radius r = 9. + ) + 5) = Center, 5), radius r =. + ) = 0 This is not a circle ) + = 5 Center 9, 0), radius r = ) 5 ) + = 0 Center 5, ), radius r = 0. + ) ) + 5 = 6 00 Center, ) 5, radius r = 6 0. ) + 5) = 65. ) + 6) = 0 5. ) + 5) = 5 6. ) + ) = ) = 096