Math Problem Statement
Solution
You have shared an image containing a problem that involves finding the equation of a line passing through the point of intersection of two given lines and the origin. Here's a breakdown of the work:
Given Lines:
The goal is to:
- Find the intersection of these two lines.
- Find the equation of the line passing through the origin and the point of intersection of and .
Step 1: Find the point of intersection of and
The parametric forms of the lines are obtained as:
Line :
where is a parameter for .
Line :
where is a parameter for .
Since the lines intersect, their coordinates must be equal at the point of intersection:
From these, we can solve for and :
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From : \lambda - \mu = \frac{3}{2} \tag{1}
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From : 2\lambda = \mu \quad \Rightarrow \quad \mu = 2\lambda \tag{2}
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Substituting into equation (1):
Now, substituting into :
Thus, the point of intersection can be found by substituting into the parametric equations of (or ):
So, the point of intersection is .
Step 2: Find the equation of the line passing through the origin and the intersection point
Now that we have the point of intersection , the equation of the line passing through the origin and this point can be written in parametric form as:
This is the equation of the line passing through the origin and the point .
Would you like a more detailed breakdown of any part of this solution? Here are some related questions for further exploration:
- How can we check if two lines are skew or intersecting?
- Can you derive the parametric form of a line given two points?
- What if the lines and were parallel? How would the solution change?
- How can we find the angle between two lines in 3D space?
- Can we express the line in vector form instead of parametric form?
Tip: When solving for points of intersection in 3D geometry, consistency between parametric equations is key to ensuring the lines truly intersect.
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Math Problem Analysis
Mathematical Concepts
Analytical Geometry
Line Intersection
Parametric Equations
Formulas
Parametric form of a line: (x - x1)/a = (y - y1)/b = (z - z1)/c
Intersection of two lines in parametric form
Theorems
Line Intersection
Suitable Grade Level
Grades 11-12
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