Find the intercepts of the circle. \[ (x+4)^{2}+(y+3)^{2}=49 \] Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The \( x \)-intercept(s) is/are (Use a comma to separate answers as needed. Type an exact answer, using radicals as needed.) There is no x-intercept.

To find the intercepts of the circle given by the equation \[ (x+4)^{2}+(y+3)^{2}=49, \] we will first identify the center and radius of the circle. The equation is in the standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. From the equation, we can see that: - The center \((h, k)\) is \((-4, -3)\). - The radius \(r\) is \(\sqrt{49} = 7\). ### Finding the \(x\)-intercepts To find the \(x\)-intercepts, we set \(y = 0\) and solve for \(x\): \[ (x + 4)^2 + (0 + 3)^2 = 49. \] This simplifies to: \[ (x + 4)^2 + 3^2 = 49, \] \[ (x + 4)^2 + 9 = 49. \] Subtracting 9 from both sides gives: \[ (x + 4)^2 = 40. \] Taking the square root of both sides, we have: \[ x + 4 = \pm \sqrt{40}. \] Since \(\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}\), we can write: \[ x + 4 = 2\sqrt{10} \quad \text{or} \quad x + 4 = -2\sqrt{10}. \] Solving for \(x\): 1. \(x + 4 = 2\sqrt{10} \Rightarrow x = 2\sqrt{10} - 4\) 2. \(x + 4 = -2\sqrt{10} \Rightarrow x = -2\sqrt{10} - 4\) Thus, the \(x\)-intercepts are: \[ x = 2\sqrt{10} - 4, \quad x = -2\sqrt{10} - 4. \] ### Finding the \(y\)-intercepts To find the \(y\)-intercepts, we set \(x = 0\) and solve for \(y\): \[ (0 + 4)^2 + (y + 3)^2 = 49. \] This simplifies to: \[ 4^2 + (y + 3)^2 = 49, \] \[ 16 + (y + 3)^2 = 49. \] Subtracting 16 from both sides gives: \[ (y + 3)^2 = 33. \] Taking the square root of both sides, we have: \[ y + 3 = \pm \sqrt{33}. \] Solving for \(y\): 1. \(y + 3 = \sqrt{33} \Rightarrow y = \sqrt{33} - 3\) 2. \(y + 3 = -\sqrt{33} \Rightarrow y = -\sqrt{33} - 3\) Thus, the \(y\)-intercepts are: \[ y = \sqrt{33} - 3, \quad y = -\sqrt{33} - 3. \] ### Summary of Intercepts - The \(x\)-intercepts are \(2\sqrt{10} - 4, -2\sqrt{10} - 4\). - The \(y\)-intercepts are \(\sqrt{33} - 3, -\sqrt{33} - 3\). ### Final Answer A. The \(x\)-intercept(s) is/are: \[ 2\sqrt{10} - 4, -2\sqrt{10} - 4. \]