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Mastering Equations of Circles: Formulas, Examples, and Graphing Tips

Learn how to write equations of circles in a coordinate plane using radius and center coordinates. Square both sides to find the standard equation. Perform exercises and graph circles efficiently with step-by-step guidance.

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Mastering Equations of Circles: Formulas, Examples, and Graphing Tips

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  1. Lesson 75 Writing the Equation of Circles

  2. You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center. Finding Equations of Circles

  3. Suppose the radius is r and the center is (h, k). Let (x, y) be any point on the circle. The distance between (x, y) and (h, k) is r, so you can use the Distance Formula. Finding Equations of Circles

  4. Finding Equations of Circles Square both sides to find the standard equation of a circle with radius r and center (h, k). (x – h)2 + (y – k)2 = r2 If the center is at the origin, then the standard equation is x2 + y2 = r2.

  5. Ex. 1: Writing a Standard Equation of a Circle Write the standard equation of the circle with a center at (-4, 1) and radius 7 (x – h)2 + (y – k)2 = r2 Standard equation of a circle. Substitute values. [(x – (-4)]2 + (y – 1)2= 72 Simplify. (x + 4)2 + (y – 1)2= 49

  6. r = r = r = r = r = Ex. 2: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. Use the Distance Formula Substitute values. Simplify. Simplify. Simplify. Simplify. r = 5

  7. Ex. 2: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. (x – h)2 + (y – k)2 = r2 Standard equation of a circle. Substitute. r = 5, (5, -1). [(x – 5)]2 + [y –(-1)]2 = 52 Simplify. (x - 5)2 + (y + 1)2 = 25

  8. Graphing Circles • If you know the equation of a circle, you can graph the circle by identifying its center and radius.

  9. Graph the circle with the equation: (x+2)2 + (y-3)2 = 9. To graph a circle you first rewrite the equation to find the center and its radius. (x+2)2 + (y-3)2 = 9 [x – (-2)]2 + (y – 3)2=32 The center is (-2, 3) and the radius is 3. Ex. 3: Graphing a circle

  10. To graph the circle, place the center point at (-2, 3), use the radius of 3 units in all directions, and draw the circle. Ex. 3: Graphing a circle

  11. Questions/Review And coming soon… Are all the same concept! Da, da, da, daaaaa!

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