Find the equation of a line passes through the point `(2,3,4)` and whose direction ratios are `3,-1,-2`.

Answer

To find the equation of a line that passes through the point \( (2, 3, 4) \) and has direction ratios \( 3, -1, -2 \), we can use the standard form of the equation of a line in three-dimensional space.

### Step-by-Step Solution:

1. **Identify the Point and Direction Ratios**:
- The point through which the line passes is \( P(2, 3, 4) \).
- The direction ratios of the line are \( L = 3 \), \( M = -1 \), and \( N = -2 \).

2. **Use the Formula for the Equation of a Line**:
- The equation of a line in three-dimensional space can be expressed as:
\[ \frac{x - x_1}{L} = \frac{y - y_1}{M} = \frac{z - z_1}{N} \]
- Here, \( (x_1, y_1, z_1) \) is the point \( (2, 3, 4) \) and \( (L, M, N) \) are the direction ratios \( (3, -1, -2) \).

3. **Substitute the Values into the Formula**:
- Substitute \( x_1 = 2 \), \( y_1 = 3 \), \( z_1 = 4 \), \( L = 3 \), \( M = -1 \), and \( N = -2 \) into the formula:
\[ \frac{x - 2}{3} = \frac{y - 3}{-1} = \frac{z - 4}{-2} \]

4. **Write the Final Equation**:
- The final equation of the line can be expressed as:
\[ \frac{x - 2}{3} = \frac{y - 3}{-1} = \frac{z - 4}{-2} \]

### Final Answer:
The equation of the line is:
\[ \frac{x - 2}{3} = \frac{y - 3}{-1} = \frac{z - 4}{-2} \]