To solve the problem, we need to find the equation of the plane that bisects the line segment joining the points A(-3, -3, 4) and B(3, 7, 6) at right angles. We will then check which of the given points lies on this plane.
### Step 1: Find the midpoint of the line segment AB
The midpoint \( C \) of the line segment joining points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is given by the formula:
\[
C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
Substituting the coordinates of points A and B:
\[
C = \left( \frac{-3 + 3}{2}, \frac{-3 + 7}{2}, \frac{4 + 6}{2} \right) = \left( \frac{0}{2}, \frac{4}{2}, \frac{10}{2} \right) = (0, 2, 5)
\]
### Step 2: Find the direction ratios of the line segment AB
The direction ratios of the line segment \( AB \) can be found using:
\[
\text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)
\]
Calculating the direction ratios:
\[
\text{Direction Ratios} = (3 - (-3), 7 - (-3), 6 - 4) = (6, 10, 2)
\]
### Step 3: Find the equation of the plane
The normal to the plane bisecting the line segment at right angles will have the same direction ratios as the line segment \( AB \). Therefore, the normal vector \( \vec{n} \) is \( (6, 10, 2) \).
Using the point-normal form of the plane equation:
\[
n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0
\]
Substituting \( \vec{n} = (6, 10, 2) \) and point \( C(0, 2, 5) \):
\[
6(x - 0) + 10(y - 2) + 2(z - 5) = 0
\]
Expanding this:
\[
6x + 10y - 20 + 2z - 10 = 0
\]
Simplifying:
\[
6x + 10y + 2z - 30 = 0
\]
Dividing the entire equation by 2:
\[
3x + 5y + z - 15 = 0
\]
### Step 4: Check which point lies on the plane
Now we need to check which of the given points satisfies the equation \( 3x + 5y + z = 15 \).
1. **Point 1: (-2, 3, 5)**
\[
3(-2) + 5(3) + 5 = -6 + 15 + 5 = 14 \quad (\text{not equal to } 15)
\]
2. **Point 2: (2, 1, 3)**
\[
3(2) + 5(1) + 3 = 6 + 5 + 3 = 14 \quad (\text{not equal to } 15)
\]
3. **Point 3: (4, 1, -2)**
\[
3(4) + 5(1) + (-2) = 12 + 5 - 2 = 15 \quad (\text{equal to } 15)
\]
4. **Point 4: (1, 2, 3)** (if applicable)
\[
3(1) + 5(2) + 3 = 3 + 10 + 3 = 16 \quad (\text{not equal to } 15)
\]
### Conclusion
The point that lies on the plane is **(4, 1, -2)**.
