Give the center and radius of the circle represented by the equation. \[ x^{2}+y^{2}+10 x+4 y+25=0 \] The center of the circle is \( \square \). (Type an ordered pair, using integers or fractions.) The radius of the circle is \( \square \). (Type an integer or a simplified fraction.)

To find the center and radius of the circle represented by the equation: \[ x^{2} + y^{2} + 10x + 4y + 25 = 0 \] **Step 1: Rewrite the Equation in Standard Form** The standard form of a circle's equation is: \[ (x - h)^{2} + (y - k)^{2} = r^{2} \] where \((h, k)\) is the center and \(r\) is the radius. **Step 2: Complete the Square for \(x\) and \(y\)** 1. **Group the \(x\) and \(y\) terms:** \[ x^{2} + 10x + y^{2} + 4y = -25 \] 2. **Complete the square for \(x\):** - Take half of the coefficient of \(x\): \(\frac{10}{2} = 5\) - Square it: \(5^{2} = 25\) - Add and subtract 25 inside the equation: \[ (x^{2} + 10x + 25) - 25 + y^{2} + 4y = -25 \] \[ (x + 5)^{2} - 25 + y^{2} + 4y = -25 \] 3. **Complete the square for \(y\):** - Take half of the coefficient of \(y\): \(\frac{4}{2} = 2\) - Square it: \(2^{2} = 4\) - Add and subtract 4 inside the equation: \[ (x + 5)^{2} - 25 + (y^{2} + 4y + 4) - 4 = -25 \] \[ (x + 5)^{2} - 25 + (y + 2)^{2} - 4 = -25 \] 4. **Combine constants:** \[ (x + 5)^{2} + (y + 2)^{2} - 29 = -25 \] \[ (x + 5)^{2} + (y + 2)^{2} = 4 \] **Step 3: Identify the Center and Radius** From the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\): - **Center \((h, k)\):** \[ (x + 5)^{2} \Rightarrow h = -5 \] \[ (y + 2)^{2} \Rightarrow k = -2 \] So, the center is \((-5, -2)\). - **Radius \(r\):** \[ r^{2} = 4 \Rightarrow r = \sqrt{4} = 2 \] **Final Answer:** - **The center of the circle is \((-5, -2)\).** - **The radius of the circle is \(2\).**