Find the intercepts of the circle. \( (x-2)^{2}+(y+7)^{2}=9 \) 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. Tyere is no \( x \)-intercept. Select the correct choice below and, if necossary, fill in the answer box within your choice. A. The \( y \)-intercept(s) is/are (Use a comma to separate answers as needed. Type an exact answer, using radicals as needed.) B. There is no \( y \)-intercept.

To find the intercepts of the circle given by the equation \((x-2)^{2}+(y+7)^{2}=9\), 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: - The center \((h, k)\) is \((2, -7)\). - The radius \(r\) is \(\sqrt{9} = 3\). ### Finding the \(x\)-intercepts: To find the \(x\)-intercepts, we set \(y = 0\) and solve for \(x\): \[ (x-2)^{2} + (0 + 7)^{2} = 9 \] \[ (x-2)^{2} + 49 = 9 \] \[ (x-2)^{2} = 9 - 49 \] \[ (x-2)^{2} = -40 \] Since the square of a real number cannot be negative, there are no \(x\)-intercepts. ### Finding the \(y\)-intercepts: To find the \(y\)-intercepts, we set \(x = 0\) and solve for \(y\): \[ (0 - 2)^{2} + (y + 7)^{2} = 9 \] \[ 4 + (y + 7)^{2} = 9 \] \[ (y + 7)^{2} = 9 - 4 \] \[ (y + 7)^{2} = 5 \] Taking the square root of both sides, we have: \[ y + 7 = \sqrt{5} \quad \text{or} \quad y + 7 = -\sqrt{5} \] Solving for \(y\): 1. \(y + 7 = \sqrt{5} \Rightarrow y = \sqrt{5} - 7\) 2. \(y + 7 = -\sqrt{5} \Rightarrow y = -\sqrt{5} - 7\) Thus, the \(y\)-intercepts are: \[ y = \sqrt{5} - 7, \quad y = -\sqrt{5} - 7 \] ### Summary of Intercepts: - The \(x\)-intercept(s) is/are: **There is no \(x\)-intercept.** - The \(y\)-intercept(s) is/are: **\(\sqrt{5} - 7, -\sqrt{5} - 7\)**.