The lines joining the points (1, 2, 3), (4, 5, 7) and (-4,3,-6), (2, 9, 2) are

Options

A. coincident
B. parallel
C. intersecting
D. None of these

Answer

To determine the relationship between the lines joining the points (1, 2, 3) to (4, 5, 7) and (-4, 3, -6) to (2, 9, 2), we need to find the direction ratios of both lines and compare them.

### Step 1: Find the direction ratios of the first line
The first line is defined by the points (1, 2, 3) and (4, 5, 7).
- The direction ratios can be calculated as:
\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the coordinates:
\[ = (4 - 1, 5 - 2, 7 - 3) = (3, 3, 4) \]

### Step 2: Find the direction ratios of the second line
The second line is defined by the points (-4, 3, -6) and (2, 9, 2).
- The direction ratios can be calculated similarly:
\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the coordinates:
\[ = (2 - (-4), 9 - 3, 2 - (-6)) = (2 + 4, 9 - 3, 2 + 6) = (6, 6, 8) \]

### Step 3: Simplify the direction ratios of the second line
To simplify the direction ratios (6, 6, 8), we can divide each component by 2:
\[ = (3, 3, 4) \]

### Step 4: Compare the direction ratios
Now we have the direction ratios of both lines:
- First line: (3, 3, 4)
- Second line: (3, 3, 4)

Since both lines have the same direction ratios, they are parallel to each other.

### Conclusion
The lines joining the points (1, 2, 3) to (4, 5, 7) and (-4, 3, -6) to (2, 9, 2) are parallel.

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