The graph of x 2 + x + y 2 + y = 199 2 in the xy -plane is

Question

The graph of x 2 + x + y 2 + y = 199 2 in the xy -plane is a circle. What is the length of the circle&rsquo;s radius?

Accepted Answer

The correct answer is 10 . It's given that the graph of x 2 + x + y 2 + y = 199 2 in the xy -plane is a circle. The equation of a circle in the xy -plane can be written in the form x - h 2 + y - k 2 = r 2 , where the coordinates of the center of the circle are h , k and the length of the radius of the circle is r . The term x - h 2 in this equation can be obtained by adding the square of half the coefficient of x to both sides of the given equation to complete the square. The coefficient of x is 1 . Half the coefficient of x is 1 2 . The square of half the coefficient of x is 1 4 . Adding 1 4 to each side of x 2 + x + y 2 + y = 199 2 yields x 2 + x + 1 4 + y 2 + y = 199 2 + 1 4 , or x + 1 2 2 + y 2 + y = 199 2 + 1 4 . Similarly, the term y - k 2 can be obtained by adding the square of half the coefficient of y to both sides of this equation, which yields x + 1 2 2 + y 2 + y + 1 4 = 199 2 + 1 4 + 1 4 , or x + 1 2 2 + y + 1 2 2 = 199 2 + 1 4 + 1 4 . This equation is equivalent to x + 1 2 2 + y + 1 2 2 = 100 , or x + 1 2 2 + y + 1 2 2 = 10 2 . Therefore, the length of the circle's radius is 10 .

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