215 2.2 Circles Starting with an equation in this general form, we can complete the square to obtain an equation of the form 1x - h22 + 1y - k22 = c, for some number c. There are three possibilities for the graph based on the value of c. 1. If c 70, then r2 = c, and the graph of the equation is a circle with radius 1c. 2. If c = 0, then the graph of the equation is the single point 1h, k2. 3. If c 60, then no points satisfy the equation, and the graph is nonexistent. The next example illustrates the procedure for finding the center and radius. General Form of the Equation of a Circle For some real numbers D, E, and F, the equation x2 +y2 +Dx +Ey +F =0 can have a graph that is a circle or a point, or is nonexistent. This form is the general form of the equation of a circle. EXAMPLE 3 Finding the Center and Radius by Completing the Square Give the center and radius of the circle with equation x2 - 6x + y2 + 10y + 18 = 0. SOLUTION We complete the square twice, once for x and once for y. Begin by subtracting 18 from each side. x2 - 6x + y2 + 10y + 18 = 0 1x2 - 6x 2 + 1y2 + 10y 2 = -18 Think: c 1 2 1-62d 2 = 1-322 = 9 and c 1 2 1102d 2 = 52 = 25 Add 9 and 25 on the left to complete the two squares, and to compensate, add 9 and 25 on the right. 1x2 - 6x + 92 + 1y2 + 10y + 252 = -18 + 9 + 25 Complete the square. 1x - 322 + 1y + 522 = 16 Factor. Add on the right. 1x - 322 + 3y - 1-5242 = 42 Center-radius form The equation represents a circle with center 13, -52 and radius 4. S Now Try Exercise 27. Add 9 and 25 on both sides.
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