The equation x 2 + y - 2 2 = 36 represents circle A. Circle B is obtained by shifting

Question

The equation x 2 + y - 2 2 = 36 represents circle A. Circle B is obtained by shifting circle A down 4 units in the xy -plane. Which of the following equations represents circle B?

Accepted Answer

Choice A is correct. The standard form of an equation of a circle in the xy -plane is x - h 2 + y - k 2 = r 2 , where the coordinates of the center of the circle are h , k and the length of the radius of the circle is r . The equation of circle A, x 2 + y - 2 2 = 36 , can be rewritten as x - 0 2 + y - 2 2 = 6 2 . Therefore, the center of circle A is at 0 , 2 and the length of the radius of circle A is 6 . If circle A is shifted down 4 units, the y -coordinate of its center will decrease by 4 ; the radius of the circle and the x -coordinate of its center will not change. Therefore, the center of circle B is at 0 , 2 - 4 , or 0 , - 2 , and its radius is 6 . Substituting 0 for h , - 2 for k , and 6 for r in the equation x - h 2 + y - k 2 = r 2 yields x - 0 2 + y - - 2 2 = 6 2 , or x 2 + y + 2 2 = 36 . Therefore, the equation x 2 + y + 2 2 = 36 represents circle B. Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, 4 units. Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, 4 units. Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, 4 units.

Question the-eq