\( 1 \leftarrow \quad \) Find the equation of the sphere passing through \( P(-2,3,4) \) and \( Q(6,-1,3) \) with its center at the midpoint of \( P Q \).

To find the equation of the sphere, we first need to determine the midpoint of points \( P(-2,3,4) \) and \( Q(6,-1,3) \). The midpoint \( M \) can be calculated as follows: \[ M = \left( \frac{-2 + 6}{2}, \frac{3 - 1}{2}, \frac{4 + 3}{2} \right) = \left( 2, 1, 3.5 \right) \] Next, we find the radius of the sphere, which is the distance from the midpoint \( M \) to either point \( P \) or \( Q \). Using the distance formula, the distance \( r \) from \( M \) to \( P \) is: \[ r = \sqrt{(2 - (-2))^2 + (1 - 3)^2 + (3.5 - 4)^2} = \sqrt{(4)^2 + (-2)^2 + (-0.5)^2} = \sqrt{16 + 4 + 0.25} = \sqrt{20.25} = 4.5 \] Now that we have the center \( M(2, 1, 3.5) \) and the radius \( r = 4.5 \), we can write the equation of the sphere: \[ (x - 2)^2 + (y - 1)^2 + \left(z - 3.5\right)^2 = (4.5)^2 \] This expands to: \[ (x - 2)^2 + (y - 1)^2 + (z - 3.5)^2 = 20.25 \] And voila! The equation of the sphere is: \[ (x - 2)^2 + (y - 1)^2 + (z - 3.5)^2 = 20.25 \]