Question

what is the center of the circle?
choose 1 answer:
a ((-2, 3))
b ((-5, 7))
c ((-4, 6))
d ((4, -6))
the circle passes through the point ((-2, 3)). what is its radius?
choose 1 answer:
a (7pi)
b (sqrt{12})
c (3.5)
d (sqrt{13})

Explanation:

First Question: What is the center of the circle?

Step1: Recall the standard form of a circle's equation: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center. But since we need to find the center from the options (assuming the circle's center is one of the given points, maybe from a diagram or context where the center is among these), we analyze the options. Wait, maybe there was a diagram, but since we have to choose, let's assume the center is \((-4, 6)\) (maybe from a typical problem setup, or maybe the distance from center to \((-2,3)\) will be calculated later for radius). Wait, actually, maybe the first question's center is C \((-4, 6)\) (let's confirm with the radius question).

Step2: For the radius, we use the distance formula between center \((h,k)\) and point \((-2,3)\). Let's take center as \((-4,6)\). Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). So \(x_1 = -4\), \(y_1 = 6\), \(x_2 = -2\), \(y_2 = 3\). Then \(d = \sqrt{(-2 - (-4))^2 + (3 - 6)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}\), which matches option D for radius. So the center is C \((-4,6)\).

Step1: Use distance formula between center \((-4,6)\) (from first question) and point \((-2,3)\). The distance formula is \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step2: Substitute \(x_1 = -4\), \(y_1 = 6\), \(x_2 = -2\), \(y_2 = 3\) into the formula: \(r = \sqrt{(-2 - (-4))^2 + (3 - 6)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}\).

Answer:

C. \((-4, 6)\)

Second Question: What is its radius?