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

The standard equation of a circle can be expressed using 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 the center \( (3, -4) \) and a radius of \( 2\sqrt{5} \), we can calculate \( r^2 \) which is \( (2\sqrt{5})^2 = 20 \). Thus, substituting the center and radius into the formula, we get \( (x - 3)^2 + (y + 4)^2 = 20 \). When tackling problems involving circles, always remember to convert the radius to its squared form to simplify calculations. Common mistakes include forgetting to follow the sign conventions for the center's coordinates or miscalculating the radius squared. Double-check your values before finalizing the equation!