If \( \mathrm{P}=(3,1) \) and \( \mathrm{Q}=(-3,-7) \), find the equation of the circle that has segment PQ as a diameter. \( (x-[?])^{2}+(y-[\quad])^{2}=[ \)

To find the equation of the circle that has segment PQ as a diameter, we first need to find the center of the circle, which is the midpoint of segment PQ. The midpoint \( M \) of segment PQ can be calculated as follows: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{3 + (-3)}{2}, \frac{1 + (-7)}{2} \right) = \left( 0, -3 \right). \] Next, we need to find the radius of the circle, which is half the distance between points P and Q. The distance \( d \) between points P and Q is given by \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{((-3) - 3)^2 + ((-7) - 1)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10. \] Thus, the radius \( r \) is \[ r = \frac{d}{2} = \frac{10}{2} = 5. \] The equation of the circle in standard form is \[ (x - h)^2 + (y - k)^2 = r^2, \] where \( (h, k) \) is the center and \( r \) is the radius. Plugging in our values, we get: \[ (x - 0)^2 + (y + 3)^2 = 5^2. \] Simplifying this gives us: \[ x^2 + (y + 3)^2 = 25. \] Thus, the final equation of the circle is: \[ \boxed{(x - 0)^2 + (y + 3)^2 = 25}. \]