Let \( (x+4)^{2}+(y-5)^{2}=40 \). Find the exact \( y \)-intercepts of the graph (if any). O There are no \( y \)-intercepts. \( \begin{array}{l}\text { O(0,5 } \pm 2 \sqrt{6}) \\ O(0,5 \pm 2 \sqrt{10}) \\ O(0,0.101) \text { and }(0,9.899) .\end{array} \)

To find the \( y \)-intercepts of the equation \( (x+4)^{2}+(y-5)^{2}=40 \), we set \( x = 0 \) and solve for \( y \): \[ (0+4)^{2}+(y-5)^{2}=40 \implies 16+(y-5)^{2}=40 \implies (y-5)^{2}=24 \] Taking the square root of both sides, we get: \[ y-5 = \pm \sqrt{24} \implies y-5 = \pm 2\sqrt{6} \implies y = 5 \pm 2\sqrt{6} \] Thus, the \( y \)-intercepts are \( (0, 5 + 2\sqrt{6}) \) and \( (0, 5 - 2\sqrt{6}) \). Many mathematical problems involving circles or ellipses can be fun! For instance, you can draw these shapes on a coordinate graph, using the center and radius to understand how they intersect with axes or other geometric figures. The critical point often lies in correctly identifying your equations! If you enjoyed working through this example, consider looking into conic sections such as parabolas, ellipses, and hyperbolas! Exploring their equations can reveal fascinating relationships in geometry and physical phenomena, such as the paths of planets or even sound waves!