Questions: The equation of a circle is given below. Identify the center and the radius. Then graph the circle. x^2+y^2-6x+2y+6=0 Center: Radius:

Solution Steps

The given equation is \(x^2 + y^2 - 6x + 2y + 6 = 0\). We rewrite this equation in the standard form of a circle, which is \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius.

To rewrite the equation in the standard form, we complete the square for the \(x\) and \(y\) terms:

\( (x^2 - 6x) + (y^2 + 2y) = -6 \) \( (x^2 - 6x + 9) + (y^2 + 2y + 1) = -6 + 9 + 1 \) \( (x - 3)^2 + (y + 1)^2 = 4 \)

Comparing the equation \((x - 3)^2 + (y + 1)^2 = 4\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\), we have: \(h = 3\), \(k = -1\), and \(r^2 = 4\). Thus, the center is \((3, -1)\) and the radius is \(r = \sqrt{4} = 2\).

To graph the circle, plot the center \((3, -1)\) on the coordinate plane. Since the radius is 2, plot points 2 units away from the center in all directions. These points would be \((5, -1)\), \((1, -1)\), \((3, 1)\), and \((3, -3)\). Draw a circle passing through these points.

Final Answer