\( 1 \leftarrow \quad \) 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 that passes through points \( P(-2,3,4) \) and \( Q(6,-1,3) \) with its center at the midpoint of \( P \) and \( Q \), we can follow these steps: 1. **Calculate the midpoint of points \( P \) and \( Q \)**: The midpoint \( M \) of two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \] 2. **Determine the radius of the sphere**: The radius \( r \) of the sphere is the distance from the center \( M \) to either point \( P \) or \( Q \). The distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] 3. **Write the equation of the sphere**: The equation of a sphere with center \( (h, k, l) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \] Now, let's perform the calculations step by step. ### Step 1: Calculate the midpoint \( M \) Using the coordinates of points \( P \) and \( Q \): \[ M = \left( \frac{-2 + 6}{2}, \frac{3 - 1}{2}, \frac{4 + 3}{2} \right) = \left( \frac{4}{2}, \frac{2}{2}, \frac{7}{2} \right) = (2, 1, 3.5) \] ### Step 2: Determine the radius \( r \) Now, we calculate the distance from \( M(2, 1, 3.5) \) to \( P(-2, 3, 4) \): \[ r = \sqrt{(2 - (-2))^2 + (1 - 3)^2 + (3.5 - 4)^2} \] Calculating each term: - \( (2 - (-2))^2 = (2 + 2)^2 = 4^2 = 16 \) - \( (1 - 3)^2 = (-2)^2 = 4 \) - \( (3.5 - 4)^2 = (-0.5)^2 = 0.25 \) Now, summing these: \[ r^2 = 16 + 4 + 0.25 = 20.25 \] ### Step 3: Write the equation of the sphere Now we can write the equation of the sphere using the center \( M(2, 1, 3.5) \) and radius \( r^2 = 20.25 \): \[ (x - 2)^2 + (y - 1)^2 + (z - 3.5)^2 = 20.25 \] Thus, the equation of the sphere is: \[ (x - 2)^2 + (y - 1)^2 + (z - 3.5)^2 = 20.25 \]