If x co-ordinate of a point on the line joining points (2,2,1) and (5,1,-2) is 4, then its z co-ordinate will be

Options

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

Answer

To find the z-coordinate of the point on the line joining the points (2, 2, 1) and (5, 1, -2) where the x-coordinate is given as 4, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**: We have two points A(2, 2, 1) and B(5, 1, -2).

2. **Use the Section Formula**:
The coordinates of a point P that divides the line segment joining points A and B in the ratio \( \lambda:1 \) can be given by:
\[ P(x, y, z) = \left( \frac{2 \cdot 1 + 5 \cdot \lambda}{\lambda + 1}, \frac{2 \cdot 1 + 1 \cdot \lambda}{\lambda + 1}, \frac{1 \cdot 1 + (-2) \cdot \lambda}{\lambda + 1} \right) \]
Here, we know that the x-coordinate of point P is 4.

3. **Set Up the Equation for x-coordinate**:
\[ \frac{2 \cdot 1 + 5 \cdot \lambda}{\lambda + 1} = 4 \]

4. **Cross Multiply**:
\[ 2 + 5\lambda = 4(\lambda + 1) \]
Simplifying this gives:
\[ 2 + 5\lambda = 4\lambda + 4 \]

5. **Rearranging the Equation**:
\[ 5\lambda - 4\lambda = 4 - 2 \]
This simplifies to:
\[ \lambda = 2 \]

6. **Find the y-coordinate** (not needed for z, but for completeness):
Using the value of \( \lambda \):
\[ y = \frac{2 \cdot 1 + 1 \cdot 2}{2 + 1} = \frac{2 + 2}{3} = \frac{4}{3} \]

7. **Find the z-coordinate**:
Using the value of \( \lambda \) in the z-coordinate formula:
\[ z = \frac{1 \cdot 1 + (-2) \cdot \lambda}{\lambda + 1} = \frac{1 - 2 \cdot 2}{2 + 1} = \frac{1 - 4}{3} = \frac{-3}{3} = -1 \]

8. **Conclusion**:
The z-coordinate of the point is \(-1\).

### Final Answer:
The z-coordinate is \(-1\).