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 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 point A be \( A(-2, 3, 4) \) and point B be \( B(6, 10, 18) \).
- We need to find a point P on the line segment AB such that the y-coordinate of P is 6.

2. **Use the Section Formula**:
The coordinates of point P that divides the line segment AB in the ratio \( m:n \) can be given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]
where \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \).

3. **Set Up the Equation for y-coordinate**:
Since the y-coordinate of point P is given as 6, we can set up the equation:
\[ \frac{m \cdot 10 + n \cdot 3}{m+n} = 6 \]
Let’s assume \( m:n = \lambda:1 \). Thus, we can rewrite the equation as:
\[ \frac{10\lambda + 3}{\lambda + 1} = 6 \]

4. **Cross Multiply to Solve for λ**:
Cross-multiplying gives:
\[ 10\lambda + 3 = 6(\lambda + 1) \]
Expanding the right side:
\[ 10\lambda + 3 = 6\lambda + 6 \]

5. **Rearranging the Equation**:
Rearranging the equation:
\[ 10\lambda - 6\lambda = 6 - 3 \]
Simplifying gives:
\[ 4\lambda = 3 \implies \lambda = \frac{3}{4} \]

6. **Finding the x and z Coordinates**:
Now we can substitute \( \lambda \) back into the section formula to find the x and z coordinates of point P:
- For the x-coordinate:
\[ x = \frac{6\lambda + (-2)}{\lambda + 1} = \frac{6 \cdot \frac{3}{4} - 2}{\frac{3}{4} + 1} = \frac{\frac{18}{4} - 2}{\frac{7}{4}} = \frac{\frac{18}{4} - \frac{8}{4}}{\frac{7}{4}} = \frac{\frac{10}{4}}{\frac{7}{4}} = \frac{10}{7} \]
- For the z-coordinate:
\[ z = \frac{18\lambda + 4}{\lambda + 1} = \frac{18 \cdot \frac{3}{4} + 4}{\frac{3}{4} + 1} = \frac{\frac{54}{4} + 4}{\frac{7}{4}} = \frac{\frac{54}{4} + \frac{16}{4}}{\frac{7}{4}} = \frac{\frac{70}{4}}{\frac{7}{4}} = 10 \]

7. **Final Coordinates of Point P**:
Thus, the coordinates of point P are:
\[ P\left(\frac{10}{7}, 6, 10\right) \]

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