The circle defined by \( x^{2}+2 a x+y^{2}+4 b y+9=0 \) has centre \( (-3 ; 2) \). (1) Determine the values of \( a \) and \( b \). (2) Calculate the radius of the circle.

To find the values of \( a \) and \( b \) for the circle defined by the equation \[ x^{2} + 2ax + y^{2} + 4by + 9 = 0, \] we can rewrite the 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. ### Step 1: Identify the center The center of the circle is given as \( (-3, 2) \). We can express the equation in a form that allows us to identify the center: 1. Rearranging the equation, we have: \[ x^2 + 2ax + y^2 + 4by = -9. \] 2. To complete the square for the \( x \) terms: \[ x^2 + 2ax = (x + a)^2 - a^2. \] 3. To complete the square for the \( y \) terms: \[ y^2 + 4by = (y + 2b)^2 - (2b)^2. \] 4. Substituting these into the equation gives: \[ (x + a)^2 - a^2 + (y + 2b)^2 - (2b)^2 = -9. \] 5. Rearranging this, we have: \[ (x + a)^2 + (y + 2b)^2 = a^2 + (2b)^2 - 9. \] ### Step 2: Set the center coordinates From the completed square form, we can identify the center: - The center \( (-3, 2) \) implies: \[ -a = -3 \quad \Rightarrow \quad a = 3, \] \[ -2b = 2 \quad \Rightarrow \quad b = -1. \] ### Step 3: Values of \( a \) and \( b \) Thus, we have: \[ a = 3, \quad b = -1. \] ### Step 4: Calculate the radius Now we substitute \( a \) and \( b \) back into the equation to find the radius: 1. Substitute \( a = 3 \) and \( b = -1 \): \[ (x + 3)^2 + (y - 2)^2 = 3^2 + (2(-1))^2 - 9. \] 2. Calculate the right side: \[ 3^2 = 9, \] \[ (2(-1))^2 = (-2)^2 = 4, \] \[ 9 + 4 - 9 = 4. \] 3. Therefore, the equation becomes: \[ (x + 3)^2 + (y - 2)^2 = 4. \] ### Step 5: Find the radius The radius \( r \) is given by: \[ r = \sqrt{4} = 2. \] ### Final Answers (1) The values of \( a \) and \( b \) are: \[ a = 3, \quad b = -1. \] (2) The radius of the circle is: \[ r = 2. \]