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10.8 Equations of Circles

10.8 Equations of Circles. Objectives. Write the equation of a circle Graph a circle on the coordinate plane using an equation. Standard Equation of a Circle. An equation for a circle with center at ( h, k ) and radius of r units is ( x – h ) 2 + ( y – k ) 2 = r 2.

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10.8 Equations of Circles

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  1. 10.8 Equations of Circles

  2. Objectives • Write the equation of a circle • Graph a circle on the coordinate plane using an equation

  3. Standard Equation of a Circle An equation for a circle with center at (h, k) and radius of r units is (x – h)2 + (y – k)2 = r 2

  4. Write an equation for a circle with the center at(3, –3),d12. Answer: Example 1a: Equation of a circle Simplify.

  5. Write an equation for a circle with the center at(–12, –1), r8. Answer: Example 1b: Equation of a circle Simplify.

  6. Write an equation for each circle. a. center at (0, –5), d18 b. center at (7, 0), r20 Answer: Answer: Your Turn:

  7. A circle with a diameter of 10 has its center in the first quadrant. The lines y–3 and x–1 are tangent to the circle. Write an equation of the circle. Example 2: Sketch a drawing of the two tangent lines.

  8. Since d10, r5. The line x–1 is perpendicular to a radius. Since x–1 is a vertical line, the radius lies on a horizontal line. Count 5 units to the right from x–1. Find the value of h. Example 2:

  9. Likewise, the radius perpendicular to the line y–3 lies on a vertical line. The value of k is 5 units up from –3. Answer: An equation for the circle is . Example 2: The center is at (4, 2), and the radius is 5.

  10. A circle with a diameter of 8 has its center in the second quadrant. The lines y–1 and x1 are tangent to the circle. Write an equation of the circle. Answer: Your Turn:

  11. Graph Example 3a: Compare each expression in the equation to the standard form. The center is at (2, –3), and the radius is 2. Graph the center. Use a compass set at a width of 2 grid squares to draw the circle.

  12. Example 3a: Answer:

  13. Graph Example 3b: Write the expression in standard form. The center is at (3, 0), and the radius is 4. Draw a circle with radius 4, centered at (3, 0).

  14. Example 3b: Answer:

  15. a. Graph Your Turn: Answer:

  16. b. Graph Your Turn: Answer:

  17. Example 4: Strategically located substations are extremely important in the transmission and distribution of a power company’s electric supply. Suppose three substations are modeled by the points D(3, 6), E(–1, 0), and F(3, –4). Determine the location of a town equidistant from all three substations, and write an equation for the circle. Explore You are given three points that lie on a circle. Plan Graph DEF. Construct the perpendicular bisectors of two sides to locate the center, which is the location of the tower. Find the length of a radius. Use the center and radius to write an equation.

  18. Example 4: Solve Graph DEF and construct the perpendicular bisectors of two sides. The center appears to be at (4, 1). This is the location of the tower. Find r by using the Distance Formula with the center and any of the three points. Write an equation.

  19. Answer: Example 4: Examine You can verify the location of the center by finding the equations of the two bisectors and solving a system of equations. You can verify the radius by finding the distance between the center and another of the three points on the circle.

  20. Answer: Your Turn: The designer of an amusement park wants to place a food court equidistant from the roller coaster located at (4, 1), the Ferris wheel located at (0, 1), and the boat ride located at (4, –3). Determine the location for the food court and write an equation for the circle.

  21. Assignment • Pre-AP GeometryPg. 578 #10 – 32 • Geometry:Pg. 578 #10 – 22, 24 - 29

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