ve the equation of a circle with a diameter that has endpoints \( (-5,6) \) and \( (2,-10) \)

To find the equation of a circle given the endpoints of its 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 endpoints \( (-5, 6) \) and \( (2, -10) \) 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 of the endpoints: \[ M = \left( \frac{-5 + 2}{2}, \frac{6 + (-10)}{2} \right) = \left( \frac{-3}{2}, \frac{-4}{2} \right) = \left( -\frac{3}{2}, -2 \right) \] So, the center of the circle is \( \left( -\frac{3}{2}, -2 \right) \). 2. **Find the radius of the circle**: The radius is half the length of the diameter. We first calculate the length of the diameter using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of the endpoints: \[ d = \sqrt{(2 - (-5))^2 + (-10 - 6)^2} = \sqrt{(2 + 5)^2 + (-10 - 6)^2} = \sqrt{7^2 + (-16)^2} = \sqrt{49 + 256} = \sqrt{305} \] The radius \( r \) is half of the diameter: \[ r = \frac{d}{2} = \frac{\sqrt{305}}{2} \] 3. **Write the equation of the circle**: The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting \( h = -\frac{3}{2} \), \( k = -2 \), and \( r = \frac{\sqrt{305}}{2} \): \[ (x + \frac{3}{2})^2 + (y + 2)^2 = \left( \frac{\sqrt{305}}{2} \right)^2 \] Simplifying the right side: \[ \left( \frac{\sqrt{305}}{2} \right)^2 = \frac{305}{4} \] Therefore, the equation of the circle is: \[ \left( x + \frac{3}{2} \right)^2 + (y + 2)^2 = \frac{305}{4} \] This is the equation of the circle with the given diameter endpoints.