Questions: The equation of a circle is given in general form. Complete parts a through d to follow the step by step process to sketch the graph of the circle. x^2+y^2+2x-6y+1=0 a) Write the equation of the circle in standard form. The equation is . (Simplify your answer.) b) Determine the center and radius of the circle. The center is (Type an ordered pair.) The radius is . (Simplify your answer. Type an exact answer, using radicals as needed.)

Solution Steps

To rewrite the given equation \(x^{2}+y^{2}+2x-6y+1=0\) in standard form, we need to complete the square for both \(x\) and \(y\).

Starting with the \(x\) terms: \[ x^2 + 2x \quad \text{can be written as} \quad (x+1)^2 - 1 \]

Next, for the \(y\) terms: \[ y^2 - 6y \quad \text{can be written as} \quad (y-3)^2 - 9 \]

Substitute these into the original equation: \[ (x+1)^2 - 1 + (y-3)^2 - 9 + 1 = 0 \]

Simplify: \[ (x+1)^2 + (y-3)^2 - 9 = 0 \] \[ (x+1)^2 + (y-3)^2 = 9 \]

From the standard form \((x+1)^2 + (y-3)^2 = 9\), we can identify the center and radius of the circle.

The center is \((-1, 3)\).

The radius is \(\sqrt{9} = 3\).

Final Answer