SECTION 1.6 Circles 53 Concepts and Vocabulary 3. Interactive Figure Exercise Exploring Circles I Open the “Circle” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Check the box “Equation of the Circle” and uncheck the boxes “Show x-intercepts” and “Show y-intercepts.” Use the sliders to set the values of h and k to 0, and the value of r to 1. What is the center of the circle? What is the radius of the circle? What is the equation of the circle? (b) Use the slider to change the value of r from 1 to 3. What is the equation of the circle whose center is the origin and radius is 3? (c) Use the slider to change the value of r from 3 to 5. What is the equation of the circle whose center is the origin and radius is 5? (d) Set the value of r to 3. Change the value of h to 2 and the value of k to 1. What is the center of the circle? What is the equation of the circle? (e) Leave the value of r set to 3 and the value of k set to 1. Change the value of h to −2. What is the center of the circle? What is the equation of the circle? (f) Leave the value of r set to 3 and the value of h set to −2. Change the value of k to −2.What is the equation of the circle? (g) Suppose the equation of a circle is ( ) ( ) − + + = x y 5 7 36 2 2 What is the center and radius of the circle? 4. Interactive Figure Exercise Exploring Circles II Open the “Circle” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Check the boxes “Equation of the Circle,” “Show x-intercepts,” and “Show y-intercepts.” Change the value of r to 3 and the value of h to 4. Set the value of k to 0. What is the center of the circle? What is the equation of the circle? Are there any y-intercepts? If so, name them. Are there any x-intercepts? If so, name them. [Note: Pay attention to the distance from the center of the circle to each coordinate axis.] (b) Leave the value of r set to 3 and the value of h set to 4. Change the value of k to −3. What is the center of the circle? What is the equation of the circle? Are there any y-intercepts? If so, name them. Are there any x-intercepts? If so, name them. [Note: Pay attention to the distance from the center of the circle to each coordinate axis.] (c) Leave the value of r set to 3 and the value of h set to 4. Change the value of k to −4.What is the equation of the circle? Are there any y-intercepts? If so, name them.Are there any x-intercepts? If so, name them. [Note: Pay attention to the distance from the center of the circle to each coordinate axis.] (d) Without graphing, how many x-intercepts will the graph of ( ) ( ) − + − = x y 5 3 9 2 2 have? (e) Without graphing, how many y-intercepts will the graph of ( ) ( ) − + − = x y 5 3 9 2 2 have? 5. True or False Every equation of the form + + + + = x y ax by c 0 2 2 has a circle as its graph. 6. For a circle, the is the distance from the center to any point on the circle. 7. True or False The radius of the circle + = x y 9 2 2 is 3. 8. True or False The circle ( ) ( ) + + − = x y 3 2 13 2 2 has center ( ) − 3, 2. 9. Multiple Choice Choose the equation of a circle with radius 6 and center ( ) − 3, 5. (a) ( ) ( ) − + + = x y 3 5 6 2 2 (b) ( ) ( ) + + − = x y 3 5 36 2 2 (c) ( ) ( ) + + − = x y 3 5 6 2 2 (d) ( ) ( ) − + + = x y 3 5 36 2 2 10. Multiple Choice The equation of a circle can be changed from general form to standard from by doing which of the following? (a) completing the squares (b) solving for x (c) solving for y (d) squaring both sides In Problems 11–14, find the center and radius of each circle. Write the standard form of the equation. Skill Building 11. x y (2, 1) (0, 1) 12. x y (1, 2) (1, 0) 13. x y (1, 2) (4, 2) 14. x y (0, 1) (2, 3) In Problems 15–26, write the standard form of the equation and the general form of the equation of each circle of radius r and center ( ) h k , . Graph each circle. 15. ( ) ( ) = = r h k 2; , 0, 0 16. ( ) ( ) = = r h k 3; , 0, 0 17. ( ) ( ) = = r h k 2; , 0, 2 18. ( ) ( ) = = r h k 3; , 1, 0 19. ( ) ( ) = = − r h k 5; , 4, 3 20. ( ) ( ) = = − r h k 4; , 2, 3 21. ( ) ( ) = = − r h k 4; , 2, 1 23. ( ) ( ) = = r h k 1 2 ; , 1 2 , 0 24. ( ) ( ) = = − r h k 1 2 ; , 0, 1 2 25. ( ) ( ) = = − r h k 13; , 5, 1 26. ( ) ( ) = = − r h k 25; , 3, 2 22. ( ) ( ) = = − − r h k 7; , 5, 2
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