15.4 Equation of a Circle

Transcription

1 Name Class Date 1.4 Equation of a Circle Essential Question: How can ou write the equation of a circle if ou know its radius and the coordinates of its center? Eplore G.1.E Show the equation of a circle with center at the origin and radius r and determine the equation for the graph of a circle with radius r and center (h, k). Deriving the Equation of a Circle Resource Locker You have alread worked with circles in several earlier lessons. Now ou will investigate circles in a coordinate plane and learn how to write an equation of a circle. We can define a circle as the set of all points in the coordinate plane that are a fied distance r from the center (h, k). Consider the circle in a coordinate plane that has its center at C (h, k) and that has radius r. Let P be an point on the circle and let the coordinates of P be (, ). Create a right triangle b drawing a horizontal line through C and a vertical line through P, as shown. What are the coordinates of point A? Eplain how ou found the coordinates of A. r P(, ) B C(h, k) A Use absolute value to write epressions for the lengths of the legs of CAP. CA = ; PA = Use the Pthagorean Theorem to write a relationship among the side lengths of CAP. + = Module 1 9 Lesson 4

2 Reflect 1. Compare our work with that of other students. Then, write an equation for the circle with center C (h, k) and radius r.. Wh do ou need absolute values when ou write epressions for the lengths of the legs in Step B, but not when ou write the relationship among the side lengths in Step C? 3. Suppose a circle has its center at the origin. What is the equation of the circle in this case? Eplain 1 Writing the Equation of a Circle You can write the equation of a circle given its coordinates and radius. Equation of a Circle The equation of a circle with center (h, k) and radius r is ( - h) + ( - k) = r. Eample 1 Write the equation of the circle with the given center and radius. Center: (-, ) ; radius: 3 ( - h) + ( - k) = r Write the general equation of a circle. ( - (-) ) + ( - ) = 3 Substitute for h, for k, and 3 for r. ( + ) + ( - ) = 9 Simplif. B Center: (4, -1) ; radius: Reflect ( - h) + ( - k) = r Write the general equation of a circle. ( - ( )) ( ) + ( - ( )) = + ( ) = Simplif. Substitute for h, for k, and for r. 4. Suppose the circle with equation ( - ) + ( + 4) = 7 is translated b (, ) ( + 3, - 1). What is the equation of the image of the circle? Eplain. Module 1 96 Lesson 4

3 Your Turn Write the equation of the circle with the given center and radius.. Center: (4, 3) ; radius: 4 6. Center: (-1, -1) ; radius: 3 Eplain Finding the Center and Radius of a Circle Sometimes ou ma find an equation of a circle in a different form. In that case, ou ma need to rewrite the equation to determine the circle s center and radius. You can use the process of completing the square to do so. Eample Find the center and radius of the circle with the given equation. Then graph the circle = Step 1 Complete the square twice to write the equation in the form ( - h) + ( - k) = r ( ) ( ) = + ( ) Set up to complete the square _-4 ( ) _ ( ) = + _-4 ( ) + _ ( ) = + Add _-4 ( ) and _ ( Simplif. ( - ) + ( + 1) = Factor. ) to both sides. Step Identif h, k, and r to determine the center and radius. h = k = -1 r = = So, the center is (, -1) and the radius is. Step 3 Graph the circle. Locate the center of the circle. Place the point of our compass at the center Open the compass to the radius. Use the compass to draw the circle. Module 1 97 Lesson 4

4 B = Step 1 Complete the square twice to write the equation in the form ( - h) + ( - k) = r ( ) ( ) = -4 + ( ) + ( ) Add ( ) and ( ) to both sides = -4 + Simplif. ( ) + ( ) = Factor. Step Identif h, k, and r to determine the center and radius. h = k = r = _ = So, the center is (, ) and the radius is. Subtract 4 from both sides. Step 3 Graph the circle. Locate the center of the circle. Place the point of our compass at the center Open the compass to the radius. Use the compass to draw the circle. Reflect 7. How can ou check our graph b testing specific points from the graph in the original equation? Give an eample. Your Turn 8. Find the center and radius of the circle with the equation =. Then graph the circle Module 1 98 Lesson 4

5 9. Find the center and radius of the circle with the equation =. Then graph the circle Eplain 3 Writing a Coordinate Proof You can use a coordinate proof to determine whether or not a given point lies on a given circle in the coordinate plane. Eample 3 Prove or disprove that the given point lies on the given circle. Point (3, _ 7 ) circle centered at the origin and containing the point (-4, ) Step 1 Plot a point at the origin and at (-4, ). Use these to help ou draw the circle centered at the origin that contains (-4, ). Step Determine the radius: r = 4 Step 3 Use the radius and the coordinates of the center to write the equation of the circle. ( - h) + ( - k) = r Write the equation of the circle. ( - ) + ( - ) = (4) + = 16 Substitute for h, for k, and 4 for r. Simplif. Step 4 Substitute the - and -coordinates of the point (3, _ 7 ) in the equation of the circle to check whether the satisf the equation of the circle. (3) + ( _ 7 ) 16 Substitute 3 for and _ 7 for = 16 Simplif. So, the point (3, _ 7 ) lies on the circle because the point s - and -coordinates satisf the equation of the circle. Module 1 99 Lesson 4

6 B Point (1, 6 ), circle with center ( 1, ) and containing the point ( 1, 3) Step 1 Plot a point at (, ) and at (-1, ). Draw the circle centered at (, ) that contains (-1, ). Step Determine the radius: r = Step 3 Use the radius and the coordinates of the center to write the equation of the circle. ( - h) + ( - k) = r Write the equation of the circle. ( - ( ) ) + ( - ( ) ) = ( ) ( ) + = Simplif. Substitute for h, for k, and for r. Step 4 Substitute the - and -coordinates of the point (1, _ 6 ) in the equation of the circle to check whether the satisf the equation of the circle. Reflect ( ) + ( ) Substitute for and for. + Simplif. So the point (1, 6 ) on the circle because the - and -coordinates of the point the equation of the circle. 1. How do ou know that the radius of the circle in Eample 3A is 4? 11. Name another point with noninteger coordinates that lies on the circle in Eample 3A. Eplain. Module 1 93 Lesson 4

7 Your Turn Prove or disprove that the given point lies on the given circle. 1. Point ( 18, -4), circle centered at the origin and containing the point (6, ) 13. Point (4, 4), circle with center (1, ) and containing the point (1, ) Elaborate 14. Discussion How is the distance formula related to the equation of a circle? 1. Essential Question Check-In What information do ou need to know to write the equation of a circle? Module Lesson 4

8 Evaluate: Homework and Practice 1. Given the equation, ( - h) = ( - k) = r, what are the coordinates of the center? Online Homework Hints and Help Etra Practice Write the equation of the circle with the given center and radius.. center: (, ) ; radius: 3. center: ( 1, 3) ; radius 8 4. center: (-4, ); radius. center: (9, ) ; radius 3 Find the center and radius of the circle with the given equation. Then graph the circle = = = = 1. Prove or disprove that the point (1, 3 ) lies on the circle that is centered at the origin and contains the point (, ). Module 1 93 Lesson 4

9 11. Prove or disprove that the point (, 3 ) lies on the circle that is centered at the origin and contains the point (-3, ). 1. Prove or disprove that the circle with equation = -3 intersects the -ais. 13. Prove or disprove that the circle with equation = -16 intersects the -ais. 14. The center of a circle is (, -8). The radius is 9. What is the equation of the circle? Select all that appl. A. + ( + 8) = 3 B. + ( + 8) = 9 C. + ( + 8) = 81 D = 17 E = Algebra Write the equation of each circle Module Lesson 4

10 17. Prove or disprove that the circle with equation = contains the point (, 4). 18. The point (, n) lies on the circle whose equation is ( - 3) + ( + ) = 6. Find the value of n. Determine whether each statement is true or false. If false, eplain wh. 19. The circle + = 7 has radius 7.. The circle ( - ) + ( + 3) = 9 passes through the point (-1, -3). 1. The center of the circle ( - 6) + ( + 4) = 1 lies in the second quadrant.. The circle ( + 1) + ( - 4) = 4 intersects the -ais. 3. The equation of the circle centered at the origin with diameter 6 is + = 36. Module Lesson 4

11 H.O.T. Focus on Higher Order Thinking 4. Multi-Step Carousels can be found in man different settings, from amusement parks to cit plazas. Suppose that the center of a carousel is at the origin and that one of the animals on the circumference of the carousel has coordinates (4, 3). a. If one unit of the coordinate plane equals 1 foot, what is the diameter of the carousel? b. As the carousel turns, the animals follow a circular path. Write the equation of this circle.. Critical Thinking The diameter of a circle has endpoints (-6, 4) and (, ). a. Write an equation for the circle in standard form. Image Credits: Lee Avison/Alam b. Prove or disprove that the point (, 4) lies on the circle. 6. Communicate Mathematical Ideas Can a unique circle be constructed from three nonlinear points? Eplain. Module 1 93 Lesson 4

12 Lesson Performance Task Cell phone towers are epensive to build, so phone companies tr to build as few towers as possible, while still ensuring that all of their customers are within range of a tower. The top figure represents the ranges of three towers that each serve customers within the shaded areas but leave customers outside the shaded areas. The bottom figure shows towers that are too close together. Customers between the towers are not left out, but man customers are served b two towers, a waste of the phone compan s mone. On a coordinate grid of three cell phone towers, each unit represents 1 mile. The graph of the range of Tower A has the equation + = 36. Tower B is 1 miles west of Tower A. Tower C is 1 miles east of Tower A. 1. Graph the ranges of the three towers. (All three have the same range.). Write the equations of the ranges of Tower B and Tower C. 3. Estimate the area of the overlap between Tower A and Tower B. 4. A new tower with a range of 8 miles is being built miles west and 11 miles north of Tower A. Write the equation of the range of the new tower. B(-1, ) A(, ) C(1, ) Module Lesson 4