Given:
Let the slopes of the two lines be m1 and m2.
It is given that:
m1 = 2m2
The tangent of the angle θ between the two lines is:
tan θ = 1/3
Step 1: Use the formula for angle between two lines
The tangent of the angle between two lines with slopes m1 and m2 is:
tan θ = |(m1 − m2) / (1 + m1m2)|
Step 2: Substitute given values
Substitute m1 = 2m2:
tan θ = |(2m2 − m2) / (1 + 2m22)|
tan θ = |m2 / (1 + 2m22)|
Given tan θ = 1/3:
|m2 / (1 + 2m22)| = 1/3
Step 3: Solve for m2
Cross-multiplying:
3|m2| = 1 + 2m22
2m22 − 3m2 + 1 = 0
Solving the quadratic equation:
2m22 − 3m2 + 1 = 0
(2m2 − 1)(m2 − 1) = 0
m2 = 1 or 1/2
Step 4: Find m1
m1 = 2m2
If m2 = 1, then m1 = 2
If m2 = 1/2, then m1 = 1
Final Answer:
The possible pairs of slopes of the two lines are:
(m1, m2) = (2, 1) or (1, 1/2)