The lines `(x)/(1)=(y)/(2)=(z)/(3)and(x-1)/(-2)=(y-2)/(-4)=(z-3)/(-6)` are

Options

A. perpendicular
B. parallel
C. intersecting
D. skew

Answer

To determine the relationship between the two given lines, we need to analyze their direction ratios and positions. Let's break down the solution step by step.

### Step 1: Identify the direction ratios of the lines

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

Thus, the direction ratios of the first line are \( (1, 2, 3) \).

The second line is given by the equation:
\[ \frac{x-1}{-2} = \frac{y-2}{-4} = \frac{z-3}{-6} \]
From this, we can extract the direction ratios of the second line:
- \( a_2 = -2 \)
- \( b_2 = -4 \)
- \( c_2 = -6 \)

To simplify, we can divide these ratios by \(-2\):
- \( a_2 = 1 \)
- \( b_2 = 2 \)
- \( c_2 = 3 \)

Thus, the direction ratios of the second line are also \( (1, 2, 3) \).

### Step 2: Compare the direction ratios

Now that we have the direction ratios for both lines:
- First line: \( (1, 2, 3) \)
- Second line: \( (1, 2, 3) \)

Since both lines have the same direction ratios, we can conclude that they are parallel.

### Step 3: Conclusion about the lines

Since the two lines are parallel, they do not intersect. Therefore, the relationship between the two lines is that they are parallel lines.

### Final Answer

The lines are **parallel**.

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