The equation x 2 + y - 1 2 = 49 represents circle A. Circle B is obtained by shifting

Question

The equation x 2 + y - 1 2 = 49 represents circle A. Circle B is obtained by shifting circle A down 2 units in the xy -plane. Which of the following equations represents circle B?

Accepted Answer

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 x 2 + y - 1 2 = 49 , which can be rewritten as x 2 + y - 1 2 = 7 2 . 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 x 2 + y + 1 2 = 7 2 , or x 2 + 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.

Question the-eq