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10.7 Warm-up

10.7 Warm-up. For each equation, make a table and then graph. Identify the y-intercept and the slope. 1. 2. 3. 4. 10.7 Warm-up (Answers). 2. For each equation, make a table and then graph . Identify each y–int . and the slope . 1. 2. 3. 4. 3. 1. 4. Geometry.

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10.7 Warm-up

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  1. 10.7 Warm-up For each equation, make a table and then graph. Identify the y-intercept and the slope. 1. 2. 3. 4. Geometry 10.7 Circles in the Coordinate Plane

  2. 10.7 Warm-up (Answers) 2 For each equation, make a table and then graph. Identify each y–int. and the slope. 1. 2. 3. 4. 3 1 4 10.7 Circles in the Coordinate Plane

  3. Geometry 10.7 Circles in the Coordinate Plane

  4. 10.7 Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? Geometry 10.7 Circles in the Coordinate Plane

  5. 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. Geometry 10.7 Circles in the Coordinate Plane

  6. Circle Definition A circle is the set of points on a plane that are equidistant from the center. The radius, r, is the distance between the center (h, k) and any point (x, y) on the circle. (x, y) r (h, k) Geometry 10.7 Circles in the Coordinate Plane

  7. Finding the Equation of a Circle Use the Distance Formula to write this. (x, y) r Square both sides: (h, k) Geometry 10.7 Circles in the Coordinate Plane

  8. The Equation of a Circle Where: (h, k) is the center r is the radius (x, y) is any point on the circle (x, y) r (h, k) Geometry 10.7 Circles in the Coordinate Plane

  9. Example 1What is the center and radius? a. (x – 9)2 + (y – 1)2 = 25 Center: (9, 1) Radius: 5 (x – 9)2 + (y – 1)2 = 52 Geometry 10.7 Circles in the Coordinate Plane

  10. Example 1What is the center and radius? b. (x – 2)2 + (y + 1)2 = 1 (x – 2)2 + (y – (-1))2 = 12 Center: (2, -1) Radius: 1 Geometry 10.7 Circles in the Coordinate Plane

  11. Example 1What is the center and radius? c. (x – 6)2 + y2 = 100 Center: (6, 0) Radius: 10 (x – 6)2 + (y – 0)2 = 102 Geometry 10.7 Circles in the Coordinate Plane

  12. Your Turn 1 Identify the center and radius of each circle: a. (x – 12)2 + (y + 3)2 = 4 Center: (12, –3) Radius = 2 b. x2+ y2 = 121 Center: (0, 0) Radius = 11 Geometry 10.7 Circles in the Coordinate Plane

  13. Example 2 Write the equation of a circle with center (5, 6) and radius 4. 42 = (x – 5)2 + (y – 6)2 16 = (x – 5)2 + (y – 6)2 or (x – 5)2 + (y – 6)2 = 16 Geometry 10.7 Circles in the Coordinate Plane

  14. Your Turn 2 Write the equation of a circle with center (1, -3) and radius 8. 82 = (x – 1)2 + (y – (-3))2 (x – 1)2 + (y + 3)2 = 64 Geometry 10.7 Circles in the Coordinate Plane

  15. What if we don’t know r? The point (3, 2) is on a circle with center (5, 4). Write the equation. Geometry 10.7 Circles in the Coordinate Plane

  16. What if we don’t know r? The point (3, 2) is on a circle with center (5, 4). Write the equation. r2 = (3 – 5)2 + (2 – 4)2 r2 = (–2 )2 + (–2)2 r2 = 4 + 4 = 8 DON’T SIMPLIFY! Geometry 10.7 Circles in the Coordinate Plane

  17. Write the equation. The point (3, 2) is on a circle with center (5, 4). Write the equation. r2 = 8 (x – 5)2 + (y – 4)2 = 8 Geometry 10.7 Circles in the Coordinate Plane

  18. Your Turn 3 The point (-1, 4) is on a circle with center (2, 3). Write the equation. r2 = (-1 – 2)2 + (4 – 3)2 r2 = (-3)2 + (1)2 r2 = 9 + 1 = 10 (x – 2)2 + (y – 3)2 = 10 Geometry 10.7 Circles in the Coordinate Plane

  19. Graphing Circles Graph the circle given by the equation (x – 2)2 + (y – 1)2 = 9 Steps: 1. Find the center (h, k). What is h? 2 • What is k? • 1 Geometry 10.7 Circles in the Coordinate Plane

  20. Graphing Circles continued (x – 2)2 + (y – 1)2 = 9 Center (2, 1) 2. Find the radius, r. 3 Why? (x – 2)2 + (y – 1)2 = 32 Geometry 10.7 Circles in the Coordinate Plane

  21. Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 3. Draw the center. 4. Draw points at the ends of 4 radii. Geometry 10.7 Circles in the Coordinate Plane

  22. Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 3. Draw the center. 4. Draw points at the ends of 4 radii. 5. Sketch the circle. Geometry 10.7 Circles in the Coordinate Plane

  23. Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 3. Draw the center. 4. Draw points at the ends of 4 radii. 5. Sketch the circle. Geometry 10.7 Circles in the Coordinate Plane

  24. Your Turn 4 Graph: (x – 1)2 + (y + 3)2 = 16 Solution: Center: (1, -3) Radius: 4 Geometry 10.7 Circles in the Coordinate Plane

  25. 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. Geometry 10.7 Circles in the Coordinate Plane

  26. Graphical Solution Graph: (x + 1)2 + (y – 1)2 = 25 Solution: Center: (-1, 1) Radius: 5 Locate (3, 4) On the circle. Geometry 10.7 Circles in the Coordinate Plane

  27. What about (3, 2)? In the interior of the circle. Geometry 10.7 Circles in the Coordinate Plane

  28. What about (-5, -3)? In the exterior of the circle. Geometry 10.7 Circles in the Coordinate Plane

  29. You could do this… Find the distance from the center (-1, 1) to the point (-5, -3): Since the distance to the point is larger than the radius, it must be in the exterior of the circle. 5 Geometry 10.7 Circles in the Coordinate Plane

  30. 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. Geometry 10.7 Circles in the Coordinate Plane

  31. Quick Practice • Identify the center and the radius of the circle: (x + 2)2 + y2 = 9 • Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle. • Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1 Geometry 10.7 Circles in the Coordinate Plane

  32. Quick Practice • Identify the center and the radius of the circle: (x + 2)2 + y2 = 9 Center (-2, 0) Radius = 3 Geometry 10.7 Circles in the Coordinate Plane

  33. Quick Practice • Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle. Geometry 10.7 Circles in the Coordinate Plane

  34. Quick Practice • Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 4 Geometry 10.7 Circles in the Coordinate Plane

  35. Homework Geometry 10.7 Circles in the Coordinate Plane

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