Question

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Explanation:

First Sub - Question: What is the center of the circle?

To find the center of a circle, we can use the visual information from the graph (even though the graph is partially shown, we can also use the fact that the center should be equidistant from all points on the circle, including the given point \((-1,-6)\) for the radius calculation later). From the options and the context, we analyze the coordinates. The center of the circle is the point that is the mid - point or the fixed point from which all points on the circle are equidistant. By looking at the options and the fact that the circle passes through \((-1,-6)\), we can see that the center is \((3,-1)\) (we can also think about the standard form of a circle \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center. When we calculate the radius later, we will confirm this).

Step 1: Recall the distance formula

The radius of a circle is the distance between the center \((h,k)\) and a point \((x,y)\) on the circle. The distance formula is \(d=\sqrt{(x - h)^2+(y - k)^2}\), where \((h,k)=(3,-1)\) (from the first sub - question) and \((x,y)=(-1,-6)\).

Step 2: Substitute the values into the distance formula

Substitute \(x=-1\), \(y = - 6\), \(h = 3\) and \(k=-1\) into the formula:

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Answer:

B. \((3,-1)\)

Second Sub - Question: The circle passes through the point \((-1,-6)\). What is its radius?