OG详解-OG4 数学2 Q25

Choice D is correct. The graph in the xy-plane of an equation of the form (x -h)² +(y -k)² = r ² is a circle with center (h, k) and a radius of length r. It’s given that circle A is represented by x² + (y -1)² = 49, which can be rewritten as x² + (y -1)² = 7². Therefore, circle A has center (0, 1) and a radius of length 7. Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point (x, y ) on circle A, the point (x, y -2) lies on circle B. Moreover, if (h, k ) is the center of circle A, then (h, k -2) is the center of circle B. Therefore, circle B has a radius of 7 and the center of circle B is (0, 1-2), or (0, -1). Thus, circle B can be represented by the equation x² +(y +1)² = 72, or x² +(y +1)² = 49.
Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right 2 units. Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up 2 units. Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left 2 units.