possible Find the equation for the circle with a diameter whose endpoints are \( (4,12) \) and \( (0,-10) \). Write the standard equation for the circle. \( \square \) (Type an equation. Type your answer in standard form. Use integers or fractions for any numbers in the equation.)

To find the equation of the circle given the endpoints of the diameter, we first need the center and the radius. 1. **Calculate the center**: The center of the circle is the midpoint of the diameter. The midpoint \( M \) between points \( (4,12) \) and \( (0,-10) \) is calculated as: \[ M = \left( \frac{4+0}{2}, \frac{12 + (-10)}{2} \right) = \left( \frac{4}{2}, \frac{2}{2} \right) = (2, 1) \] 2. **Calculate the radius**: The radius \( r \) is half the distance of the diameter. The distance \( d \) of the endpoints is calculated using the distance formula: \[ d = \sqrt{(4-0)^2 + (12 - (-10))^2} = \sqrt{(4)^2 + (12 + 10)^2} = \sqrt{16 + 484} = \sqrt{500} = 10\sqrt{5} \] Thus, the radius \( r \) is: \[ r = \frac{d}{2} = \frac{10\sqrt{5}}{2} = 5\sqrt{5} \] 3. **Write the standard equation**: The standard form of the equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. Substituting in our values: \[ (x - 2)^2 + (y - 1)^2 = (5\sqrt{5})^2 \] Simplifying, we have: \[ (x - 2)^2 + (y - 1)^2 = 125 \] Thus, the standard equation for the circle is: \[ (x - 2)^2 + (y - 1)^2 = 125 \]