OG详解-OG10 数学1 Q23

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 the y-axis at exactly one point, then the perpendicular distance from the center of the circle to this point on the y-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 the y-axis at exactly one point.  
Choice A is incorrect. This is the equation of a circle that does not intersect the y-axis at any point. Choice B is incorrect. This is an equation of a circle that intersects the x-axis, not the y-axis, at exactly one point. Choice D is incorrect. This is the equation of a circle with the center located on the y-axis and thus intersects the y-axis at exactly two points, not exactly one point.