https://file.aiquickdraw.com/mathbot/file/1728128294296gjiynlyj.jpg

The problem shown in the image involves finding the equation of a line passing through the point of intersection of two lines \( L_1 \) and \( L_2 \), and the origin.

Here’s a breakdown of the steps in the image:

1. **Equation of Line 1 (\( L_1 \)):**
   \[
   \frac{x - 1}{2} = \frac{y + 3}{4} = \frac{z - 5}{3}
   \]
   This is the parametric form of the line \( L_1 \).

2. **Equation of Line 2 (\( L_2 \)):**
   \[
   \frac{x - 4}{2} = \frac{y + 3}{2} = \frac{z - 4}{4}
   \]
   This is the parametric form of the line \( L_2 \).

3. **Parametric Coordinates for Points on \( L_1 \) and \( L_2 \):**
   - For \( L_1 \), the point on the line is represented as:
     \[
     (2\lambda + 1, 4\lambda - 3, 3\lambda + 5)
     \]
   - For \( L_2 \), the point is represented as:
     \[
     (2\mu + 4, 2\mu - 3, 4\mu + 4)
     \]

4. **Intersection Condition:**
   As the two lines intersect, we set the parametric equations equal to each other, solving for the values of \( \lambda \) and \( \mu \) at the intersection point. This part seems to involve matching terms like:
   \[
   4\lambda - 3 = 2\mu - 3
   \]
   \[
   3\lambda + 5 = 4\mu + 4
   \]
   These will give the values of \( \lambda \) and \( \mu \), from which the coordinates of the intersection point can be found.

5. **Final Step (Not Visible):**
   After finding the intersection point, the next step would be to use this point and the origin to find the equation of the line passing through both the intersection point and the origin.

Would you like more detailed steps on solving this system and finding the final line equation? Let me know if you'd like to clarify any part of the process.

### Here are 5 follow-up questions for further exploration:
1. How do we find the exact values of \( \lambda \) and \( \mu \) for the intersection point of the lines?
2. What is the parametric form of a line passing through two points?
3. How can we verify if two lines in 3D space are parallel, intersecting, or skew?
4. How do parametric equations help in solving 3D geometry problems?
5. What happens if two lines in space are skew—how would we approach finding the shortest distance between them?

### Tip:
When solving parametric equations for intersecting lines, remember to match the coordinates of \( x \), \( y \), and \( z \) separately to solve for the parameters.

Find the equation of the line passing through the intersection point of two lines, L1 and L2, and the origin.

Learn how to find the equation of a line passing through the intersection point of two 3D parametric lines, L1 and L2, and the origin. This problem involves solving parametric equations and understanding line intersection in 3D geometry. Suitable for students in Grades 10-12.

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