Trite the standard form of the equation of the circle with endpoints of a diameter at the points \( (9,6) \) and \( (-7,9) \). The standard form of the equation of the circle is Type an equation. Simplify your answer.)

To find the standard form of the equation of a circle given the endpoints of a diameter, we first need to determine the center and the radius of the circle. 1. **Find the center of the circle**: The center of the circle is the midpoint of the diameter. The midpoint \( M \) of the points \( (x_1, y_1) = (9, 6) \) and \( (x_2, y_2) = (-7, 9) \) can be calculated using the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates: \[ M = \left( \frac{9 + (-7)}{2}, \frac{6 + 9}{2} \right) = \left( \frac{2}{2}, \frac{15}{2} \right) = (1, 7.5) \] So, the center of the circle is \( (1, 7.5) \). 2. **Find the radius of the circle**: The radius is half the distance between the endpoints of the diameter. We can find the distance \( d \) between the points \( (9, 6) \) and \( (-7, 9) \) using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates: \[ d = \sqrt{((-7) - 9)^2 + (9 - 6)^2} = \sqrt{(-16)^2 + (3)^2} = \sqrt{256 + 9} = \sqrt{265} \] The radius \( r \) is half of this distance: \[ r = \frac{d}{2} = \frac{\sqrt{265}}{2} \] 3. **Write the standard form of the equation of the circle**: The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting \( h = 1 \), \( k = 7.5 \), and \( r = \frac{\sqrt{265}}{2} \): \[ (x - 1)^2 + \left(y - 7.5\right)^2 = \left(\frac{\sqrt{265}}{2}\right)^2 \] Calculating \( r^2 \): \[ \left(\frac{\sqrt{265}}{2}\right)^2 = \frac{265}{4} \] Thus, the equation of the circle is: \[ (x - 1)^2 + (y - 7.5)^2 = \frac{265}{4} \] This is the standard form of the equation of the circle.