The co-ordinates of the point which divides the line joining `(2, 3, 4)` and `(3,-4,7)` in the ratio 2:4 are :

Options

A. (10,1,1)
B. (1,10,1)
C. (10,-10,10)
D. (1,1,10)

Answer

To find the coordinates of the point that divides the line segment joining the points \( A(2, 3, 4) \) and \( B(3, -4, 7) \) in the ratio \( 2:4 \), we can use the section formula. The section formula in three-dimensional geometry states that if a point \( C(x, y, z) \) divides the line segment joining points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of point \( C \) can be calculated as follows:

\[ C\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B**:
- \( A(2, 3, 4) \) and \( B(3, -4, 7) \)
- Here, \( x_1 = 2, y_1 = 3, z_1 = 4 \) and \( x_2 = 3, y_2 = -4, z_2 = 7 \).

2. **Identify the ratio**:
- The ratio in which the point divides the line segment is \( m:n = 2:4 \).
- Here, \( m = 2 \) and \( n = 4 \).

3. **Calculate the coordinates of point C using the section formula**:
- For the x-coordinate:
\[ x = \frac{mx_2 + nx_1}{m+n} = \frac{2 \cdot 3 + 4 \cdot 2}{2 + 4} = \frac{6 + 8}{6} = \frac{14}{6} = \frac{7}{3} \]

- For the y-coordinate:
\[ y = \frac{my_2 + ny_1}{m+n} = \frac{2 \cdot (-4) + 4 \cdot 3}{2 + 4} = \frac{-8 + 12}{6} = \frac{4}{6} = \frac{2}{3} \]

- For the z-coordinate:
\[ z = \frac{mz_2 + nz_1}{m+n} = \frac{2 \cdot 7 + 4 \cdot 4}{2 + 4} = \frac{14 + 16}{6} = \frac{30}{6} = 5 \]

4. **Combine the coordinates**:
- The coordinates of point \( C \) are:
\[ C\left( \frac{7}{3}, \frac{2}{3}, 5 \right) \]

### Final Answer:
The coordinates of the point which divides the line joining \( (2, 3, 4) \) and \( (3, -4, 7) \) in the ratio \( 2:4 \) are \( \left( \frac{7}{3}, \frac{2}{3}, 5 \right) \).