which equations represent circles that have a diameter of 12 units and a center that lies on the y - axis…

which equations represent circles that have a diameter of 12 units and a center that lies on the y - axis? select two options.\n$x^{2}+(y - 3)^{2}=36$\n$x^{2}+(y - 5)^{2}=6$\n$(x - 4)^{2}+y^{2}=36$\n$(x + 6)^{2}+y^{2}=144$\n$x^{2}+(y + 8)^{2}=36$

which equations represent circles that have a diameter of 12 units and a center that lies on the y - axis? select two options.\n$x^{2}+(y - 3)^{2}=36$\n$x^{2}+(y - 5)^{2}=6$\n$(x - 4)^{2}+y^{2}=36$\n$(x + 6)^{2}+y^{2}=144$\n$x^{2}+(y + 8)^{2}=36$

Answer

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is ((x - h)^2+(y - k)^2=r^2), where ((h,k)) is the center and (r) is the radius. Given diameter (d = 12), then radius (r=\frac{d}{2}=6) (so (r^{2}=36)). Also, since the center lies on the (y) - axis, (h = 0).

Step2: Analyze each option

  • For (x^{2}+(y - 3)^{2}=36): Here (h = 0,k = 3,r^{2}=36). The center ((0,3)) lies on the (y) - axis and (r = 6) (since (r^{2}=36)).
  • For (x^{2}+(y - 5)^{2}=6): Here (r^{2}=6\neq36), so this is incorrect.
  • For ((x - 4)^{2}+y^{2}=36): Here (h = 4\neq0), so the center ((4,0)) does not lie on the (y) - axis.
  • For ((x + 6)^{2}+y^{2}=144): Here (r^{2}=144), so (r = 12\neq6).
  • For (x^{2}+(y + 8)^{2}=36): Here (h = 0,k=-8,r^{2}=36). The center ((0, - 8)) lies on the (y) - axis and (r = 6) (since (r^{2}=36)).

Answer:

(x^{2}+(y - 3)^{2}=36), (x^{2}+(y + 8)^{2}=36)