The lines `(x-2)/(1)=(y-3)/(2)=(z-4)/(3)and(x-1)/(-5)=(y-2)/(1)=(z-1)/(1)` are

Options

A. parallel
B. at right angle
C. intersecting
D. skew lines

Answer

To determine the relationship between the given lines, we will follow these steps:

### Step 1: Identify the Direction Ratios of the Lines
The first line is given by:
\[ \frac{x-2}{1} = \frac{y-3}{2} = \frac{z-4}{3} \]
From this, we can extract the direction ratios as:
- \( a_1 = 1 \)
- \( b_1 = 2 \)
- \( c_1 = 3 \)

The second line is given by:
\[ \frac{x-1}{-5} = \frac{y-2}{1} = \frac{z-1}{1} \]
From this, we can extract the direction ratios as:
- \( a_2 = -5 \)
- \( b_2 = 1 \)
- \( c_2 = 1 \)

### Step 2: Check if the Lines are Parallel
For two lines to be parallel, their direction ratios must be proportional. We will check if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
Calculating the ratios:
\[ \frac{1}{-5}, \quad \frac{2}{1}, \quad \frac{3}{1} \]
These ratios are not equal, therefore the lines are not parallel.

### Step 3: Check if the Lines are Perpendicular
To check if the lines are perpendicular, we will calculate the dot product of their direction ratios. The dot product is given by:
\[ a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 \]
Substituting the values:
\[ 1 \cdot (-5) + 2 \cdot 1 + 3 \cdot 1 = -5 + 2 + 3 = 0 \]
Since the dot product is zero, the lines are perpendicular.

### Conclusion
The lines are neither parallel nor intersecting at an angle other than 90 degrees. Therefore, the final conclusion is that the lines are perpendicular.