The straight line `(x-3)/2=(y-4)/3=(z-5)/4` is parallel to which one of the following?

Options

A. `4x+3y-5z=0`
B. `4x+5y-4z=0`
C. `4x+4y-5z=0`
D. `5x+4y-5z=0`

Explanation

To determine which line is parallel to the given line \((x-3)/2 = (y-4)/3 = (z-5)/4\), we need to find the direction ratios (DRs) of the given line and compare them with the direction ratios of the other lines provided in the options.

### Step-by-Step Solution:

1. **Identify the Direction Ratios of the Given Line**:
The given line can be expressed in the symmetric form:
\[ \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4} \]
From this, we can identify the direction ratios (DRs) of the line as:
\[ (2, 3, 4) \]

2. **Understand the Condition for Parallelism**:
Two lines are parallel if their direction ratios are proportional. For a line with direction ratios \( (a, b, c) \) to be parallel to another line with direction ratios \( (a', b', c') \), the following condition must hold:
\[ a \cdot a' + b \cdot b' + c \cdot c' = 0 \]
Here, \( (a, b, c) \) are the direction ratios of the given line, and \( (a', b', c') \) are the direction ratios of the line we are checking.

3. **Check Each Option**:
We will check each option provided to see if the condition for parallelism holds.

- **Option 1**: Suppose the direction ratios are \( (4, 3, -5) \).
\[ 2 \cdot 4 + 3 \cdot 3 + 4 \cdot (-5) = 8 + 9 - 20 = -3 \quad (\text{Not parallel}) \]

- **Option 2**: Suppose the direction ratios are \( (4, 5, -4) \).
\[ 2 \cdot 4 + 3 \cdot 5 + 4 \cdot (-4) = 8 + 15 - 16 = 7 \quad (\text{Not parallel}) \]

- **Option 3**: Suppose the direction ratios are \( (4, 4, -5) \).
\[ 2 \cdot 4 + 3 \cdot 4 + 4 \cdot (-5) = 8 + 12 - 20 = 0 \quad (\text{Parallel}) \]

- **Option 4**: Check if there is another option, but we already found the parallel line in option 3.

4. **Conclusion**:
The line \((x-3)/2 = (y-4)/3 = (z-5)/4\) is parallel to the line represented by option 3, with direction ratios \( (4, 4, -5) \).

### Final Answer:
The straight line \((x-3)/2 = (y-4)/3 = (z-5)/4\) is parallel to the line with direction ratios \( (4, 4, -5) \).