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Circles: Objectives/Assignment

Circles: Objectives/Assignment. Write the equation of a circle. Use the equation of a circle and its graph to solve problems. Graphing a circle using its four quick points. CIRCLES. What do you know about circles?. Definitions.

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Circles: Objectives/Assignment

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  1. Circles:Objectives/Assignment • Write the equation of a circle. • Use the equation of a circle and its graph to solve problems. • Graphing a circle using its four quick points.

  2. CIRCLES What do you know about circles?

  3. Definitions • Circle:The set of all points that are the same distance (equidistant) from a fixed point. • Center: the fixed points • Radius:a segment whose endpoints are the center and a point on the circle

  4. The equation of circle centered at (0,0) and with radius r Solution: Let P(x, y) represent any point on the circle x 2 + y 2 = r 2

  5. Finding the Equation of a Circle The center is (0, 0) The radius is 12 The equation is: x 2 + y 2 = 144

  6. Write out the equation for a circle centered at (0, 0) with radius =1 Solution: Let P(x, y) represent any point on the circle

  7. Equation of a Circle

  8. Ex. 1: Writing a Standard Equation of a Circle centered at (-4, 0) and radius 7.1 (x – h)2 + (y – k)2 = r2 Standard equation of a circle. [(x – (-4)]2 + (y – 0)2 =7.12Substitute values. (x + 4)2 + y2 = 50.41 Simplify.

  9. Ex. 2: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5,-1). Use the Distance Formula Substitute values. Simplify Addition Property Square root the result.

  10. The equation of the circle is: • (x –5)2 +[y –(-1)]2 = 52 • (x –5)2 +(y + 1)2 = 25 Simplify

  11. You try!!: Write the equation of the circle : (x-5)2 + ( y+4)2 = 49 (x+3)2 + ( y- 6)2 = 25

  12. THINK ABOUT IT Find the center, the length of the radius, and write the equation of the circle if the endpoints of a diameter are (-8,2) and (2,0). Center: Use midpoint formula! Length: use distance formula with radius and an endpoint Equation: Put it all together

  13. Finding the Equation of a Circle Circle A The center is (16, 10) The radius is 10 The equation is (x – 16)2 + (y – 10)2 = 100

  14. Finding the Equation of a Circle Circle B The center is (4, 20) The radius is 10 The equation is (x – 4)2 + (y – 20)2 = 100

  15. Finding the Equation of a Circle Circle O The center is (0, 0) The radius is 12 The equation is x 2 + y 2 = 144

  16. Graphing Circles • If you know the equation of a circle, you can graph the circle • by identifying its center and radius; • By listing four quick points: the upmost, lowest, leftmost and rightmost points.

  17. Graphing Circles Using 4 quick points (x – 3)2 + (y – 2)2 = 9 Center (3, 2) Radius of 3 Leftmost point (0, 2) Rightmost point(6, 2) Highest point(3, 5) Lowest point(3, -1)

  18. Graphing Circles by listing its four quick points (x + 4)2 + (y – 1)2 = 25 Center (-4, 1) Radius of 5 Left: (-9, 1) Right: (1, 1) Up: (-4, 6) Low: (-4, -4)

  19. Ex. 4: Applying Graphs of Circles A bank of lights is arranged over a stage. Eachlight illuminates a circular area on the stage. A coordinate plane is used to arrange the lights, using the corner of the stage as the origin. The equation (x – 13)2 + (y - 4)2 = 16 represents one of the disks of light. A. Graph the disk of light. B. Three actors are located as follows: Henry is at (11, 4), Jolene is at (8, 5), and Martin is at (15, 5). Which actors are in the disk of light?

  20. Ex. 4: Applying Graphs of Circles • 1. Rewrite the equation to find the center and • radius. • – (x – h)2 + (y – k)2= r2 • – (x - 13)2 + (y - 4)2 = 16 • – (x – 13)2 + (y – 4)2= 42 • – The center is at (13, 4) and the radius is 4. The • circle is shown on the next slide.

  21. Ex. 4: Applying Graphs of Circles Graph the disk of light The graph shows that Henry and Martin are both in the disk of light.

  22. Ex. 4: Applying Graphs of Circles A bank of lights is arranged over a stage. Each light is

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