Questions: Session 2025 Question 5 of 40 Find the center and the radius of the circle with the given equation. Then draw the graph. x^2+y^2-6x+2y=71 The center of the circle is (Type an ordered pair.) The radius of the circle is Use the graphing tool to graph the circle.

Solution Steps

The given equation of the circle is:

\[ x^{2} + y^{2} - 6x + 2y = 71 \]

To find the center and radius, we need to rewrite this equation in the standard form of a circle, \((x - h)^2 + (y - k)^2 = r^2\).

First, complete the square for the \(x\) terms and the \(y\) terms.

For the \(x\) terms:

\[ x^2 - 6x \quad \Rightarrow \quad (x - 3)^2 - 9 \]

For the \(y\) terms:

\[ y^2 + 2y \quad \Rightarrow \quad (y + 1)^2 - 1 \]

Substitute the completed squares back into the equation:

\[ (x - 3)^2 - 9 + (y + 1)^2 - 1 = 71 \]

Simplify:

\[ (x - 3)^2 + (y + 1)^2 = 81 \]

The equation \((x - 3)^2 + (y + 1)^2 = 81\) is in the standard form \((x - h)^2 + (y - k)^2 = r^2\).

• The center \((h, k)\) is \((3, -1)\).
• The radius \(r\) is \(\sqrt{81} = 9\).

Final Answer