OG详解-OG2 数学2 Q24

Choice C is correct. The graph of the equation  (x -h)² + (y -k)²  = r²   in the xy-plane is a circle with center  (h, k) and a radius of length  r. The radius of a    circle is the distance from the center of the circle to any point on the circle. If a circle in the xy-plane intersects they-axis at exactly one point, then the perpendicular distance from the center of the circle to this point on they-axis must be equal to the length of the circle’s radius. It follows that the x-coordinate of the circle’s center must be equivalent to the length of the circle’s radius. In other words, if the graph of (x -h)² + (y -k)²  = r²   is a circle that intersects the    y-axis at exactly one point, then  r = h  must be true. The equation in choice C is (x -4)2 + (y -9)2  = 16 , or (x -4)2 + (y -9)2  = 42 . This equation is in the form (x -h)² + (y -k)²  = r², where h = 4, k = 9, and r = 4, and represents a circle in the xy-plane with center (4,9) and radius of length  4. Substituting  4  for r  and  4  for h in the equation r = ︳h ︳yields 4 =  ︳4 ︳, or 4 = 4, which is true. Therefore, the equation in choice C represents a circle in the xy-plane that intersects they-axis at exactly one point.
Choice A is incorrect. This is the equation of a circle that does not intersect they-axis at any point. Choice B is incorrect. This is an equation of a circle that intersects the x-axis, not they-axis, at exactly one point. Choice D is incorrect. This is the equation of a circle with the center located on they-axis and thus intersects they-axis at exactly two points, not exactly one point.