OG详解-OG3 数学1 Q27

The correct answer is 10. It’s given that the graph of x² + x +y² +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 = r², 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 14​​. Adding 1​4​  to each side of (x² + x)+ (y²+y)= 199​2​ yields ( x² + x +1​4​ ）+  (y² + y) =  199​2​+1​4​ , or (x+12​)²+(y²+y)=1992​+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+12​)²+(y²+y+14​)=1992​+14​+1​4​  or  (x+12​)²+(y2+12​)2​=1992​+14​+14​.  This equation is equivalent to (x+12​)²+(y+12​)²=100​  or  (x+12​)2+(y+12​)2=10^2​ . Therefore, the length of the circle’sradius is 10.