To determine the relationship between the two given lines in three-dimensional geometry, we will analyze their direction ratios and check if they are parallel, coincident, or skew.
### Step 1: Write down the equations of the lines
The first line is given by:
\[
\frac{x-2}{1} = \frac{y+4}{2} = \frac{z-3}{3}
\]
The second line is given by:
\[
\frac{x}{2} = \frac{y-1}{4} = \frac{z+3}{6}
\]
### Step 2: Identify the direction ratios of both lines
For the first line, the direction ratios can be extracted as follows:
- From \(\frac{x-2}{1}\), we have \(l_1 = 1\)
- From \(\frac{y+4}{2}\), we have \(m_1 = 2\)
- From \(\frac{z-3}{3}\), we have \(n_1 = 3\)
Thus, the direction ratios of the first line are \( (1, 2, 3) \).
For the second line:
- From \(\frac{x}{2}\), we have \(l_2 = 2\)
- From \(\frac{y-1}{4}\), we have \(m_2 = 4\)
- From \(\frac{z+3}{6}\), we have \(n_2 = 6\)
Thus, the direction ratios of the second line are \( (2, 4, 6) \).
### Step 3: Check the proportionality of the direction ratios
Now we check if the direction ratios are proportional:
\[
\frac{l_1}{l_2} = \frac{1}{2}, \quad \frac{m_1}{m_2} = \frac{2}{4} = \frac{1}{2}, \quad \frac{n_1}{n_2} = \frac{3}{6} = \frac{1}{2}
\]
Since all the ratios are equal, we conclude that the direction ratios are proportional. Therefore, the two lines are either parallel or coincident.
### Step 4: Find a point on the second line
To determine if the lines are coincident or merely parallel, we need to find a point on the second line:
From the second line's equation, we can set:
\[
x = 0 \implies \frac{0}{2} = \frac{y-1}{4} = \frac{z+3}{6}
\]
This gives us:
- For \(y\):
\[
\frac{0}{2} = \frac{y-1}{4} \implies y - 1 = 0 \implies y = 1
\]
- For \(z\):
\[
\frac{0}{2} = \frac{z+3}{6} \implies z + 3 = 0 \implies z = -3
\]
Thus, a point on the second line is \( (0, 1, -3) \).
### Step 5: Check if this point lies on the first line
Now we need to check if the point \( (0, 1, -3) \) lies on the first line:
Substituting \(x = 0\), \(y = 1\), and \(z = -3\) into the first line's equations:
1. For \(x\):
\[
\frac{0-2}{1} = -2
\]
2. For \(y\):
\[
\frac{1+4}{2} = \frac{5}{2}
\]
3. For \(z\):
\[
\frac{-3-3}{3} = -2
\]
### Step 6: Compare the values of \(\lambda\)
We find:
- From \(x\): \(\lambda = -2\)
- From \(y\): \(\lambda = \frac{5}{2}\)
- From \(z\): \(\lambda = -2\)
Since the values of \(\lambda\) are not the same for all three coordinates, the point \( (0, 1, -3) \) does not lie on the first line.
### Conclusion
Since the lines are parallel (as their direction ratios are proportional) but do not intersect (the point from the second line does not lie on the first line), we conclude that the two lines are parallel.
### Final Answer
The two lines are **parallel**.
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