Questions: Which of the following is true about a circle whose equation is (x+5)^2+(y-3)^2=36? (1) It has a center of (5,-3) and an area of 12 pi. (2) It has a center of (-5,3) and a diameter of 6. (3) It has a center of (-5,3) and an area of 36 pi. (4) It has a center of (5,-3) and a circumference of 12 pi.

Solution Steps

The equation of the circle is given in the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\), where \((h, k)\) is the center and \(r\) is the radius.
For the equation \((x+5)^{2} + (y-3)^{2} = 36\), the center is \((-5, 3)\).

The right-hand side of the equation, \(36\), represents \(r^{2}\).
Thus, the radius \(r = \sqrt{36} = 6\).
The area of a circle is given by \(A = \pi r^{2}\). Substituting \(r = 6\), the area is \(A = \pi (6)^{2} = 36\pi\).

• Option (1): Claims the center is \((5, -3)\) and the area is \(12\pi\). This is incorrect because the center is \((-5, 3)\) and the area is \(36\pi\).
• Option (2): Claims the center is \((-5, 3)\) and the diameter is 6. This is incorrect because the diameter is \(2r = 12\), not 6.
• Option (3): Claims the center is \((-5, 3)\) and the area is \(36\pi\). This is correct.
• Option (4): Claims the center is \((5, -3)\) and the circumference is \(12\pi\). This is incorrect because the center is \((-5, 3)\) and the circumference is \(2\pi r = 12\pi\), but the center is wrong.

The correct option is (3).

Final Answer