Find the standard form of the equation of the circle with endpoints of a diameter at the points \( (9,2) \) and \( (-1,8) \). Type the standard form of the equation of this circle. (Type an equation.)

To find the standard form of the equation of a circle, we need to determine the center and the radius. The standard form of the equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center of the circle and \(r\) is the radius. Given the endpoints of the diameter are \((9,2)\) and \((-1,8)\), we can find the center by averaging the x-coordinates and the y-coordinates of the endpoints: \[ h = \frac{x_1 + x_2}{2} = \frac{9 + (-1)}{2} = \frac{8}{2} = 4 \] \[ k = \frac{y_1 + y_2}{2} = \frac{2 + 8}{2} = \frac{10}{2} = 5 \] So the center of the circle is \((4, 5)\). Next, we need to find the radius. The radius is the distance from the center to either endpoint of the diameter. We can use the distance formula to find the radius: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Using the coordinates of the endpoints \((9,2)\) and \((-1,8)\): \[ r = \sqrt{(-1 - 9)^2 + (8 - 2)^2} = \sqrt{(-10)^2 + (6)^2} = \sqrt{100 + 36} = \sqrt{136} \] Now we have the center \((4, 5)\) and the radius \(\sqrt{136}\). We can write the standard form of the equation of the circle: \[ (x - 4)^2 + (y - 5)^2 = (\sqrt{136})^2 \] \[ (x - 4)^2 + (y - 5)^2 = 136 \] So the standard form of the equation of the circle with endpoints of a diameter at the points \((9,2)\) and \((-1,8)\) is: \[ (x - 4)^2 + (y - 5)^2 = 136 \]