If the x-coordinate of a point `P` on the join of `Q(2,2,1)` and `R(5,1,-2)` is 4, then its z-coordinate is

Options

A. 2
B. 1
C. -1
D. -2

Explanation

To find the z-coordinate of point \( P \) on the line segment joining points \( Q(2, 2, 1) \) and \( R(5, 1, -2) \) when the x-coordinate of \( P \) is given as 4, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the coordinates of points Q and R**:
- \( Q(2, 2, 1) \)
- \( R(5, 1, -2) \)

2. **Let the coordinates of point P be \( P(x, y, z) \)**. Since we know the x-coordinate of \( P \) is 4, we can write:
- \( P(4, y, z) \)

3. **Use the section formula**:
The coordinates of point \( P \) dividing the line segment \( QR \) in the ratio \( \lambda:1 \) are given by:
\[ P_x = \frac{5\lambda + 2}{\lambda + 1}, \quad P_y = \frac{1\lambda + 2}{\lambda + 1}, \quad P_z = \frac{-2\lambda + 1}{\lambda + 1} \]

4. **Set up the equation for the x-coordinate**:
Since \( P_x = 4 \):
\[ \frac{5\lambda + 2}{\lambda + 1} = 4 \]

5. **Cross-multiply to solve for \( \lambda \)**:
\[ 5\lambda + 2 = 4(\lambda + 1) \]
Expanding the right side gives:
\[ 5\lambda + 2 = 4\lambda + 4 \]

6. **Rearranging the equation**:
\[ 5\lambda - 4\lambda = 4 - 2 \]
\[ \lambda = 2 \]

7. **Substitute \( \lambda \) into the formula for \( z \)**:
Using the value of \( \lambda \) in the formula for \( P_z \):
\[ P_z = \frac{-2\lambda + 1}{\lambda + 1} = \frac{-2(2) + 1}{2 + 1} \]
Simplifying this gives:
\[ P_z = \frac{-4 + 1}{3} = \frac{-3}{3} = -1 \]

8. **Final answer**:
The z-coordinate of point \( P \) is \( -1 \).

### Summary:
The z-coordinate of point \( P \) is \( -1 \).