213 2.2 Circles 2.2 Circles ■ Center-Radius Form ■ General Form ■ An Application Center-Radius Form By definition, a circle is the set of all points in a plane that lie a given distance from a given point. The given distance is the radius of the circle, and the given point is the center. We can find the equation of a circle from its definition using the distance formula. Suppose that the point 1h, k2 is the center and the circle has radius r, where r 70. Let 1x, y2 represent any point on the circle. See Figure 13. 0 r h k x y (x, y) (h, k) Figure 13 21x2 - x122 + 1y 2 - y122 = d Distance formula 21x - h22 + 1y - k22 = r 1x 1, y12 = 1h, k2, 1x2, y22 = 1x, y2, and d = r 1 x −h22 + 1 y −k22 =r2 Square each side. LOOKING AHEAD TO CALCULUS The circle x2 +y2 =1 is called the unit circle. It is important in interpreting the trigonometric or circular functions that appear in the study of calculus. Center-Radius Form of the Equation of a Circle A circle with center 1h, k2 and radius r has equation 1 x −h22 + 1 y −k22 =r2, which is the center-radius form of the equation of the circle. As a special case, a circle with center 10, 02 and radius r has the following equation. x2 +y2 =r2 Be careful with signs here. EXAMPLE 1 Finding the Center-Radius Form Find the center-radius form of the equation of each circle described. (a) center 1-3, 42, radius 6 (b) center 10, 02, radius 3 SOLUTION (a) 1x - h22 + 1y - k22 = r2 Center-radius form 3x - 1-3242 + 1y - 422 = 62 Substitute. Let 1h, k2 = 1-3, 42 and r = 6. 1x + 322 + 1y - 422 = 36 Simplify. (b) The center is the origin and r = 3. x2 + y2 = r2 Special case of the center-radius form x2 + y2 = 32 Let r = 3. x2 + y2 = 9 Apply the exponent. S Now Try Exercises 11(a) and 17(a).
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