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Write equations and graph circles in the coordinate plane. Objectives

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The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center.

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Example 1A: Writing the Equation of a Circle Write the equation of each circle. J with center J (2, 2) and radius 4 (x – h) 2 + (y – k) 2 = r 2 (x – 2) 2 + (y – 2) 2 = 4 2 (x – 2) 2 + (y – 2) 2 = 16 Equation of a circle Substitute 2 for h, 2 for k, and 4 for r. Simplify.

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Example 1B: Writing the Equation of a Circle Write the equation of each circle. K that passes through J(6, 4) and has center K(1, –8) Distance formula. Simplify. (x – 1) 2 + (y – (–8)) 2 = 13 2 (x – 1) 2 + (y + 8) 2 = 169 Substitute 1 for h, –8 for k, and 13 for r. Simplify.

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Write the equation of each circle. P with center P(0, –3) and radius 8 (x – h) 2 + (y – k) 2 = r 2 (x – 0) 2 + (y – (–3)) 2 = 8 2 x 2 + (y + 3) 2 = 64 Equation of a circle Substitute 0 for h, –3 for k, and 8 for r. Simplify. Example 1C: Writing the Equation of a Circle

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If you are given the equation of a circle, you can graph the circle by identifying its center and radius.

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Example 2A: Graphing a Circle Graph x 2 + y 2 = 16. Step 1: Find the radius. Since the radius is, or 4, use ±4. Step 2: Find the center (h, k) The center is (0, 0).

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Example 2B: Graphing a Circle Graph (x – 3) 2 + (y + 4) 2 = 9. The equation of the given circle can be written as (x – 3) 2 + (y – (– 4)) 2 = 3 2. So h = 3, k = –4, and r = 3. The center is (3, –4). Step 1: Find the radius. Step 2: Find the center (h, k) = 3

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