What is the standard equation of the circle whose center is \( (3,-4) \) and radus is \( 2 \sqrt{5} \) ? \( \begin{array}{l}O(x-3)^{2}+(y-4)^{2}=20 \\ (x-3)^{2}+(y+4)^{2}=10 \\ O(x-3)^{2}+(y+4)^{2}=20 \\ O(x+3)^{2}+(y-4)^{2}=20\end{array} \)

To find the standard equation of a circle, we use the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center of the circle and \(r\) is the radius. Given: - Center \((h, k) = (3, -4)\) - Radius \(r = 2\sqrt{5}\) First, we need to calculate \(r^2\): \[ r^2 = (2\sqrt{5})^2 = 4 \cdot 5 = 20 \] Now, substituting the values into the circle equation: \[ (x - 3)^2 + (y + 4)^2 = 20 \] Thus, the standard equation of the circle is: \[ (x - 3)^2 + (y + 4)^2 = 20 \] Now, let's compare this with the options provided: 1. \(O(x-3)^{2}+(y-4)^{2}=20\) 2. \((x-3)^{2}+(y+4)^{2}=10\) 3. \(O(x-3)^{2}+(y+4)^{2}=20\) 4. \(O(x+3)^{2}+(y-4)^{2}=20\) The correct equation is: \[ (x - 3)^2 + (y + 4)^2 = 20 \] This matches option 3 (with the "O" being a typographical error). Therefore, the answer is: \[ (x - 3)^2 + (y + 4)^2 = 20 \]