Lines

Step 1: Parameterize the two given lines
The parametric equations of the lines are: so any point on is: For the second line: so any point on is: Step 2: Find the point of intersection of the two lines
Equating the coordinates of and : From , . Substitute into the other equations: Thus, the point of intersection is: Step 3: Find the direction ratios of the required line
The direction ratios of the given lines are: The direction ratios of the line perpendicular to both and are given by: Calculate the cross product: Step 4: Write the equation of the required line
The equation of the required line passing through with direction ratios is:

1. Point of intersection of the two lines:
Let: From the first line: From the second line: Equate for consistency:
- From :
- From : Substitute into the first line: Thus, the point of intersection is .
2. Direction vectors of the given lines:
- First line: .
- Second line: .
3. Direction of the required line:
The required line is perpendicular to both and . Use the cross product: Expand: 4. Equation of the required line:
The line passing through with direction vector is: Final Answer:

Step 1: Recall the direction cosine condition
For a line making angles with the positive directions of the -axis, -axis, and -axis respectively, the sum of the squares of the direction cosines is:
Step 2: Substitute the given angles
The line makes angles and with the -axis and -axis, so: Since , we have: Simplify:
Step 3: Solve for
Combine terms:
Thus:
Step 4: {Conclude the result}
The angle which the line makes with the positive direction of the -axis is .

To solve the problem, we are given the parametric equations of a line:
, , and
We need to check which of the given points satisfies all three equations for some value of .

1. Check Option (A): (1, -5, 6)
Try substituting into the equations:
- From , putting :

- Check ✔️
- Check ❌ (Expected z = 6, but we get -6)
So, (A) is not on the line.

2. Check Option (B): (1, 5, 6)
-
- ❌ (Expected y = 5)
So, (B) is not on the line.

3. Check Option (C): (1, -5, -6)
-
- ✔️
- ✔️
All three coordinates satisfy the line’s equations when .

4. Conclusion:
The point (1, -5, -6) lies on the line.

Final Answer:
The correct option is (C) (1, -5, -6).