A point with y-coordinate 6 lies on the line segment
joining the points (-2, 3, 4) and (6, 10, 18).
Coordinates of the point are

Options

A. `(10/7, -6, 10)`
B. `(-10/7, 6, -10)`
C. `(10/7, 6, 10)`
D. (0, 0, 0)

Answer

To find the coordinates of the point \( C \) with a y-coordinate of 6 that lies on the line segment joining the points \( A(-2, 3, 4) \) and \( B(6, 10, 18) \), we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A(-2, 3, 4) \) and \( B(6, 10, 18) \).
- The point \( C \) has coordinates \( (x, 6, z) \).

2. **Use the Section Formula**:
- The section formula states that if a point \( C \) divides the line segment \( AB \) in the ratio \( k:1 \), then the coordinates of \( C \) can be calculated as:
\[ C\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]
- Here, \( (x_1, y_1, z_1) = (-2, 3, 4) \) and \( (x_2, y_2, z_2) = (6, 10, 18) \).

3. **Set Up the Equation for the y-coordinate**:
- Since the y-coordinate of point \( C \) is given as 6, we can set up the equation using the y-coordinates of points \( A \) and \( B \):
\[ \frac{10k + 3}{k + 1} = 6 \]

4. **Solve for \( k \)**:
- Cross-multiply to eliminate the fraction:
\[ 10k + 3 = 6(k + 1) \]
- Expanding the right side:
\[ 10k + 3 = 6k + 6 \]
- Rearranging gives:
\[ 10k - 6k = 6 - 3 \]
\[ 4k = 3 \quad \Rightarrow \quad k = \frac{3}{4} \]

5. **Calculate the x-coordinate**:
- Substitute \( k \) into the x-coordinate formula:
\[ x = \frac{6 \cdot k + (-2) \cdot 1}{k + 1} = \frac{6 \cdot \frac{3}{4} + (-2) \cdot 1}{\frac{3}{4} + 1} \]
- Simplifying:
\[ x = \frac{\frac{18}{4} - 2}{\frac{3}{4} + \frac{4}{4}} = \frac{\frac{18}{4} - \frac{8}{4}}{\frac{7}{4}} = \frac{\frac{10}{4}}{\frac{7}{4}} = \frac{10}{7} \]

6. **Calculate the z-coordinate**:
- Substitute \( k \) into the z-coordinate formula:
\[ z = \frac{18 \cdot k + 4 \cdot 1}{k + 1} = \frac{18 \cdot \frac{3}{4} + 4}{\frac{3}{4} + 1} \]
- Simplifying:
\[ z = \frac{\frac{54}{4} + \frac{16}{4}}{\frac{7}{4}} = \frac{\frac{70}{4}}{\frac{7}{4}} = 10 \]

7. **Final Coordinates**:
- The coordinates of point \( C \) are:
\[ C\left(\frac{10}{7}, 6, 10\right) \]

### Conclusion:
The coordinates of the point \( C \) are \( \left(\frac{10}{7}, 6, 10\right) \).