4.1 Circles. Deriving the Standard-Form Equation of a Circle. Explore

Transcription

1 Name Class Date 4.1 Circles ssential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? plore Deriving the Standard-Form quation of a Circle Recall that a circle is the set of points in a plane that are a fied distance, called the radius, from a given point, called the center. Resource Locker The coordinate plane shows a circle with center C(h, k) and radius r. P(, ) is an arbitrar point on the circle but is not directl above or below or to the left or right of C. (, k) is a point with the same -coordinate as P and the same -coordinate as C. plain wh CP is a right triangle. r P (, ) C (h, k) (, k) Houghton Mifflin Harcourt Publishing Compan C Identif the lengths of the sides of CP. Remember that point P is arbitrar, so ou cannot rel upon the diagram to know whether the -coordinate of P is greater than or less than h or whether the -coordinate of P is greater than or less than k, so ou must use absolute value for the lengths of the legs of CP. lso, remember that the length of the hpotenuse of CP is just the radius of the circle. The length of segment C is. The length of segment P is. The length of segment CP is. ppl the Pthagorean Theorem to CP to obtain an equation of the circle. ( - ) + ( - ) = Module Lesson 1

2 Reflect 1. Discussion Wh isn t absolute value used in the equation of the circle?. Discussion Wh does the equation of the circle also appl to the cases in which P has the same -coordinate as C or the same -coordinate as C so that CP doesn t eist? plain 1 Writing the quation of a Circle The standard-form equation of a circle with center C(h, k) and radius r is ( - h) + ( - k) = r. If ou solve this equation for r, ou obtain the equation r = ( -h) + ( - k), which gives ou a means for finding the radius of a circle when the center and a point P(, ) on the circle are known. ample 1 Write the equation of the circle. The circle with center C(-3, ) and radius r = 4 Substitute -3 for h, for k, and 4 for r into the general equation and simplif. ( - (-3) ) + ( - ) = 4 ( + 3) + ( - ) = 16 The circle with center C(, -3) and containing the point P(, 5) Step 1 Find the radius. r = CP ( - ()) = ( ) = = + = _ = + ( ) + ( - (-3)) Houghton Mifflin Harcourt Publishing Compan Step Write the equation of the circle. ( - ()) + ( - (-3)) = ( + 4) + ( + 3) = Module Lesson 1

3 Your Turn Write the equation of the circle. 3. The circle with center C(1, ) and radius r = 4. The circle with center C(-, 5) and containing the point P(-, -1) plain Rewriting an quation of a Circle to Graph the Circle panding the standard-form equation ( - h) + ( - k) = r results in a general second-degree equation in two variables having the form + + c + d + e = 0. In order to graph such an equation or an even more general equation of the form a + a + c + d + e = 0. ou must complete the square on both and to put the equation in standard form and identif the circle s center and radius. ample Graph the circle after writing the equation in standard form = 0 Write the equation = 0 Prepare to complete the square on and. ( ) + ( ) = Complete both squares. ( ) + ( ) = Houghton Mifflin Harcourt Publishing Compan Factor and simplif. ( - 5) + ( + 3) = 4 The center of the circle is C(5, -3), and the radius is r = _ 4 =. Graph the circle Module Lesson 1

4 = 0 Write the equation = 0 Factor 4 from the terms 4 ( + ) + 4 ( - 4) + 11 = 0 and the terms. Prepare to complete 4 ( + + ) + 4 ( ) = ( ) + 4 ( ) the square on and. Complete both 4 ( + + ) + 4 ( ) = ( ) + 4 ( ) squares. Factor and simplif. 4 ( + ) + 4 ( - ) = Divide both sides b 4. ( + ) + ( - ) = The center is C (, ), and the radius is r = _ =. Graph the circle Your Turn Graph the circle after writing the equation in standard form = = Houghton Mifflin Harcourt Publishing Compan Module 4 16 Lesson 1

5 plain 3 Solving a Real-World Problem Involving a Circle circle in a coordinate plane divides the plane into two regions: points inside the circle and points outside the circle. Points inside the circle satisf the inequalit ( - h) + ( - k) < r, while points outside the circle satisf the inequalit ( - h) + ( - k) > r. ample 3 Write an inequalit representing the given situation, and draw a circle to solve the problem. The table lists the locations of the homes of five friends along with the locations of their favorite pizza restaurant and the school the attend. The friends are deciding where to have a pizza part based on the fact that the restaurant offers free deliver to locations within a 3-mile radius of the restaurant. t which homes should the friends hold their pizza part to get free deliver? Houghton Mifflin Harcourt Publishing Compan Image Credits: Jose Luis Pelaez/Corbis Place Location lonzo s home (3, ) arbara s home (, 4) Constance s home C(-, 3) Dion s home D(0, -1) li s home (1, ) Pizza restaurant (-1, 1) School (1, -) Write the equation of the circle with center (-1, 1) and radius 3. ( - (-1)) + ( - 1) = 3, or ( + 1) + ( - 1) = 9 The inequalit ( + 1) + ( - 1) < 9 represents the situation. Plot the points from the table and graph the circle. The points inside the circle satisf the inequalit. So, the friends should hold their pizza part at either Constance s home or Dion s home to get free deliver. In order for a student to ride the bus to school, the student must live more than miles from the school. Which of the five friends are eligible to ride the bus? Write the equation of the circle with center (, ) and radius. + ( - ( )) = ( - ) ( - ) + ( + ) The inequalit ( - ) Use the coordinate grid in Part to graph the circle. = + ( + ) > represents the situation. The points the circle satisf the inequalit. So, are eligible to ride the bus. 4 C Restaurant - 0 D 4 - School Module Lesson 1

6 Reflect 7. For Part, how do ou know that point isn t outside the circle? Your Turn Write an inequalit representing the given situation, and draw a circle to solve the problem. 8. Sasha delivers newspapers to subscribers that live within a 4-block radius of her house. Sasha s house is located at point (0, -1). Points,, C, D, and represent the houses of some of the subscribers to the newspaper. To which houses does Sasha deliver newspapers? C Sasha - D laborate 9. Describe the process for deriving the equation of a circle given the coordinates of its center and its radius. 10. What must ou do with the equation a + a + c + d + e = 0 in order to graph it? 11. What do the inequalities ( - h) + ( - k) < r and ( - h) + ( - k) > r represent? 1. ssential Question Check-In What information must ou know or determine in order to write an equation of a circle in standard form? Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

7 valuate: Homework and Practice Write the equation of the circle. 1. The circle with C (4, -11) and radius r = 16 Online Homework Hints and Help tra Practice. The circle with C (-7, -1) and radius r = The circle with center C (-8, ) and containing the point P (-1, 6) 4. The circle with center C (5, 9) and containing the point P (4, 8) Houghton Mifflin Harcourt Publishing Compan In ercises 5 1, graph the circle after writing the equation in standard form = Module Lesson 1

8 Graph the circle after writing the equation in standard form = = = = Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

9 = = = Houghton Mifflin Harcourt Publishing Compan In ercises 13 0, write an inequalit representing the problem, and draw a circle to solve the problem. 13. router for a wireless network on a floor of an office building has a range of 35 feet. The router is located at the point (30, 30). The lettered points in the coordinate diagram represent computers in the office. Which computers will be able to connect to the network through the router? C D F Router G Module Lesson 1

10 Write an inequalit representing the problem, and draw a circle to solve the problem. 14. The epicenter of an earthquake is located at the point (0, -30). The earthquake is felt up to 40 miles awa. The labeled points in the coordinate diagram represent towns near the epicenter. In which towns is the earthquake felt? C picenter F D 15. ida s cat has disappeared somewhere in her apartment. The last time she saw the cat, it was located at the point (30, 40). ida knows all of the cat s hiding places, which are indicated b the lettered points in the coordinate diagram. If she searches for the cat no farther than 5 feet from where she last saw it, which hiding places will she check? D F G 0 C rock concert is held in a large state park. The concert stage is located at the point (-, ), and the music can be heard as far as 4 miles awa. The lettered points in the coordinate diagram represent campsites within the park. t which campsites can the music be heard? 8 C H 4 D F G L -8 K J Houghton Mifflin Harcourt Publishing Compan Image Credits wonderlandstock/lam: Module Lesson 1

11 17. usiness When Claire started her in-home computer service and support business, she decided not to accept clients located more than 10 miles from her home. Claire s home is located at the point (5, 0), and the lettered points in the coordinate diagram represent the homes of her prospective clients. Which prospective clients will Claire not accept? -0 0 F C D G viation n airport s radar sstem detects airplanes that are in flight as far as 60 miles from the airport. The airport is located at (-0, 40). The lettered points in the coordinate diagram represent the locations of airplanes currentl in flight. Which airplanes does the airport s radar sstem detect? 80 J G D F C K -80 H Houghton Mifflin Harcourt Publishing Compan Image Credits Mikael Damkier/Shutterstock 19. Due to a radiation leak at a nuclear power plant, the towns up to a distance of 30 miles from the plant are to be evacuated. The nuclear power plant is located at the point (-10, -10). The lettered points in the coordinate diagram represent the towns in the area. Which towns are in the evacuation zone? 0 D F C 0 Module Lesson 1

12 0. ats that live in a cave at point (-10, 0) have a feeding range of 40 miles. The lettered points in the coordinate diagram represent towns near the cave. In which towns are bats from the cave not likel to be observed? Write an inequalit representing the problem, and draw a circle to solve the problem. 0 D 40 C 0 G F Match the equations to the center and radius of the circle each represents. Show our work = 0 C (9, -11) ; r = = 0 C (9, 11) ; r = 15 C = 0 C (-9, -11) ; r = 15 D = 0 C (-9, 11) ; r = 13 Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

13 H.O.T. Focus on Higher Order Thinking. Multi-Step garden sprinkler waters the plants in a garden within a 1-foot spra radius. The sprinkler is located at the point (5, -10). The lettered points in the coordinate diagram represent the plants. Use the diagram for parts a c. a. Write an inequalit that represents the region that does not get water from the sprinkler. Then draw a circle and use it to identif the plants that do not get water from the sprinkler G C F -10 D -0 b. Suppose a second sprinkler with the same spra radius is placed at the point (10, 10). Write a sstem of inequalities that represents the region that does not get water from either sprinkler. Then draw a second circle and use it to identif the plants that do not get water from either sprinkler. Houghton Mifflin Harcourt Publishing Compan c. Where would ou place a third sprinkler with the same spra radius so all the plants get water from a sprinkler? Write a sstem of inequalities that represents the region that does not get water from an of the sprinklers. Then draw a third circle to show that ever plant receives water from a sprinkler. Module Lesson 1

14 3. Represent Real-World Situations The orbit of the planet Venus is nearl circular. n astronomer develops a model for the orbit in which the Sun has coordinates S(0, 0), the circular orbit of Venus passes through V (41, 53), and each unit of the coordinate plane represents 1 million miles. Write an equation for the orbit of Venus. How far is Venus from the sun? 4. Draw Conclusions The unit circle is defined as the circle with radius 1 centered at the origin. Pthagorean triple is an ordered triple of three positive integers, (a, b, c), that satisf the relationship a + b = c. n eample of a Pthagorean triple is (3, 4, 5). In parts a d, ou will draw conclusions about Pthagorean triples. a. Write the equation of the unit circle. b. Use the Pthagorean triple (3, 4, 5) and the smmetr of a circle to identif the coordinates of two points on the part of the unit circle that lies in Quadrant I. plain our reasoning. Houghton Mifflin Harcourt Publishing Compan Image Credits: Digital Vision/Gett Images Module 4 17 Lesson 1

15 c. Use our answer from part b and the smmetr of a circle to identif the coordinates of si other points on the unit circle. This time, the points should be in Quadrants II, III, and IV. d. Find a different Pthagorean triple and use it to identif the coordinates of eight points on the unit circle. 5. Make a Conjecture In a two-dimensional plane, coordinates are given b ordered pairs of the form (, ). You can generalize coordinates to three-dimensional space b using ordered triples of the form (,, z) where the coordinate z is used to indicate displacement above or below the -plane. Generalize the standard-form equation of a circle to find the general equation of a sphere. plain our reasoning. Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

16 Lesson Performance Task highwa that runs straight east and west passes 6 miles south of a radio tower. The broadcast range of the station is 10 miles. N a. Determine the distance along the highwa that a car will be within range of the radio station s signal. b. Given that the car is traveling at a constant speed of 60 miles per hour, determine the amount of time the car is within range of the signal. 10 miles 6 miles Radio tower Houghton Mifflin Harcourt Publishing Compan Module Lesson 1