SECTION 1.6 Circles 51 Look back at Figure 74 or 75 on the previous page to verify the approximate locations of the intercepts. Now Work PROBLEM 29(C) 3 Work with the General Form of the Equation of a Circle If we eliminate the parentheses from the standard form of the equation of the circle given in Example 2, we get ( ) ( ) + + − = + + + − + = x y x x y y 3 2 16 6 9 4 4 16 2 2 2 2 which simplifies to + + − − = x y x y 6 4 3 0 2 2 It can be shown that any equation of the form + + + + = x y ax by c 0 2 2 has a graph that is a circle, is a point, or has no graph at all. For example, the graph of the equation + = x y 0 2 2 is the single point ( ) 0, 0 . The equation + + = x y 5 0, 2 2 or + = − x y 5, 2 2 has no graph, because sums of squares of real numbers are never negative. DEFINITION General Form of the Equation of a Circle When its graph is a circle, the equation + + + + = x y ax by c 0 2 2 is the general form of the equation of a circle . If an equation of a circle is in general form, we use the method of completing the square to put the equation in standard form so that we can identify its center and radius. Need to Review? Completing the square is discussed in Section A.3, p. A29 . Graphing a Circle Whose Equation Is in General Form Graph the equation: + + − + = x y x y 4 6 12 0 2 2 Solution EXAMPLE 4 Group the terms involving x, group the terms involving y, and put the constant on the right side of the equation. The result is ( ) ( ) + + − = − x x y y 4 6 12 2 2 Next, complete the square of each expression in parentheses. Remember that any number added on the left side of the equation must also be added on the right. Factor. x x y y x y 4 4 6 9 12 4 9 4 2 4 6 2 9 2 3 1 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) + + + − + = − + + = ↑ − = ↑ + + − = This equation is the standard form of the equation of a circle with radius 1 and center ( ) −2, 3 . To graph the equation by hand, use the center ( ) −2, 3 and the radius 1. See Figure 76(a) on the next page. To graph the equation using some graphing utilities, solve for y. y x y x y x 3 1 2 3 1 2 3 1 2 2 2 2 2 ( ) ( ) ( ) ( ) − = − + − = ± − + = ± − + Use the Square Root Method. Add 3 to both sides. (continued)
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