A circle in the xy -plane has its center at - 4 , 5 and the point - 8 ,

Question

A circle in the xy -plane has its center at - 4 , 5 and the point - 8 , 8 lies on the circle. Which equation represents this circle?

Accepted Answer

Choice D is correct. A circle in the xy -plane can be represented by an equation of the form x - h 2 + y - k 2 = r 2 , where h , k is the center of the circle and r is the length of a radius of the circle. It's given that the circle has its center at - 4 , 5 . Therefore, h = - 4 and k = 5 . Substituting - 4 for h and 5 for k in the equation x - h 2 + y - k 2 = r 2 yields x - - 4 2 + y - 5 2 = r 2 , or x + 4 2 + y - 5 2 = r 2 . It's also given that the point - 8 , 8 lies on the circle. Substituting - 8 for x and 8 for y in the equation x + 4 2 + y - 5 2 = r 2 yields - 8 + 4 2 + 8 - 5 2 = r 2 , or - 4 2 + 3 2 = r 2 , which is equivalent to 16 + 9 = r 2 , or 25 = r 2 . Substituting 25 for r 2 in the equation x + 4 2 + y - 5 2 = r 2 yields x + 4 2 + y - 5 2 = 25 . Thus, the equation x + 4 2 + y - 5 2 = 25 represents the circle. Choice A is incorrect. The circle represented by this equation has its center at 4 , - 5 , not - 4 , 5 , and the point - 8 , 8 doesn't lie on the circle. Choice B is incorrect. The point - 8 , 8 doesn't lie on the circle represented by this equation. Choice C is incorrect. The circle represented by this equation has its center at 4 , - 5 , not - 4 , 5 , and the point - 8 , 8 doesn't lie on the circle.

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