Geometry 10.6 Equations of a Circle

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles2 Goals Write the equation of a circle. Use the equation of a circle to graph the circle on the coordinate plane. Solve problems with circles.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles3 Circle Definition A circle is the set of points on a plane that are equidistant from the center. r (x, y) (h, k) The radius, r, is the distance between the center (h, k) and any point (x, y) on the circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles4 Circle Equation r (x, y) (h, k) Use the Distance Formula to write this.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles5 Circle Equation r (x, y) (h, k) Square both sides:

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles6 The Equation of a Circle r (x, y) (h, k) Where: (h, k) is the center r is the radius (x, y) is any point on the circle

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles7 What is the center and radius? (x – 9) 2 + (y – 1) 2 = 25 Center: (9, 1) Radius: 5 (x – 9) 2 + (y – 1) 2 = 5 2

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles8 What is the center and radius? (x – 2) 2 + (y + 1) 2 = 1 (x – 2) 2 + (y – (-1)) 2 = 1 2 Center: (2, -1) Radius: 1

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles9 What is the center and radius? (x – 6) 2 + y 2 = 100 Center: (6, 0) Radius: 10 (x – 6) 2 + (y – 0) 2 = 10 2

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles10 Your Turn Identify the center and radius of each circle: (x – 12) 2 + (y + 3) 2 = 4 Center: (12, –3) Radius = 2 x 2 + y 2 = 121 Center: (0, 0) Radius = 11

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles11 Example Write the equation of a circle with center (5, 6) and radius = (x – 5) 2 + (y – 6) 2 16 = (x – 5) 2 + (y – 6) 2 or (x – 5) 2 + (y – 6) 2 = 16

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles12 Your Turn Write the equation of a circle with center (1, -3) and radius = (x – 1) 2 + (y – (-3)) 2 (x – 1) 2 + (y + 3) 2 = 64

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles13 What if we dont know r? The point (3, 2) is on a circle with center (5, 4). Write the equation.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles14 What if we dont know r? The point (3, 2) is on a circle with center (5, 4). Write the equation. r 2 = (3 – 5) 2 + (2 – 4) 2 r 2 = (–2 ) 2 + (–2) 2 r 2 = = 8 DONT SIMPLIFY!

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles15 Write the equation. The point (3, 2) is on a circle with center (5, 4). Write the equation. r 2 = 8 (x – 5) 2 + (y – 4) 2 = 8

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles16 Your Turn. The point (-1, 4) is on a circle with center (2, 3). Write the equation. r 2 = (-1 – 2) 2 + (4 – 3) 2 r 2 = (-3) 2 + (1) 2 r 2 = = 10 (x – 2) 2 + (y – 3) 2 = 10

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles17 Graphing Circles Graph the circle given by the equation (x – 2) 2 + (y – 1) 2 = 9 First find the center (h, k). What is h? 2 What is k? 1

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles18 Graphing Circles continued (x – 2) 2 + (y – 1) 2 = 9 Center (2, 1) What is r? 3 Why? (x – 2) 2 + (y – 1) 2 = 3 2

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles19 Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 1)Draw the center. 2)Draw points at the ends of 4 radii.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles20 Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 1)Draw the center. 2)Draw points at the ends of 4 radii. 3)Sketch the circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles21 Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 1)Draw the center. 2)Draw points at the ends of 4 radii. 3)Sketch the circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles22 Your Turn Graph: (x – 1) 2 + (y + 3) 2 = 16 Solution: Center: (1, -3) Radius: 4

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles23 Problem (x + 1) 2 + (y – 1) 2 = 25 Is the point (3, 4) on the circle, in its interior, or in the exterior? Directions: Make a sketch of the circle. Then locate (3, 4) and answer the question.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles24 Graphical Solution Graph: (x + 1) 2 + (y – 1) 2 = 25 Solution: Center: (-1, 1) Radius: 5 Locate (3, 4) On the circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles25 What about (3, 2)? In the interior of the circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles26 What about (-5, -3)? In the exterior of the circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles27 You could do this… Since the distance to the point is larger than the radius, it must be in the exterior of the circle. Find the distance from the center (-1, 1) to the point (-5, -3): 5

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles28 What you can now do: Write the equation of a circle. Graph a circle from its equation. Determine where a point is in the interior, exterior, or on a circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles29 Quick Practice 1.Identify the center and the radius of the circle: (x + 2) 2 + y 2 = 9 2.Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle. 3.Sketch the graph of the circle given by the equation (x - 1) 2 + (y + 3) 2 = 1

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles30 Quick Practice 1.Identify the center and the radius of the circle: (x + 2) 2 + y 2 = 9 Center (-2, 0) Radius = 3

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles31 Quick Practice 2.Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle.

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles32 Quick Practice 3.Sketch the graph of the circle given by the equation (x - 1) 2 + (y + 3) 2 = 1

Thursday, March 26, 2:44 Geometry 10.6 Equations of Circles33 Homework