The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3, find the slopes of the lines

Solution:

We know that if θ is the angle between the lines l₁ and l₂ with slopes m₁ and m₂ then

tanθ = |(m₂ - m₁)/(1 + m₁m₂)|

It is given that the tangent of the angle between the two lines is 1/3 and slope of a line is double of the slope of another line.

Let m and 2m be the slopes of the given lines.

Therefore,

1/3 = |(2m - m)/(1 + m(2m)|

1/3 = |(m)/(1 + 2m²)|

1/3 = (m)/(1 + 2m²) (or) 1/3 = (- m)/(1 + 2m²)

Case I:

1/3 = (- m)/(1 + 2m²)

1+ 2m² = -3m

2m² + 3m + 1 = 0

2m² + 2m + m + 1 = 0

2m (m + 1) + 1(m + 1) = 0

(m + 1)(2m + 1) = 0

⇒ m = - 1 or m = - 1/2

If m = - 1, then the slopes of the lines are - 1 and - 2.

If m = - 1/2, then the slopes of the lines are - 1/2 and - 1.

Case II:

1/3 = - (m)/(1 + 2m²)

2m² + 1 = 3m

2m² - 3m + 1 = 0

2m² - 2m - m + 1 = 0

2m (m - 1) - 1(m - 1) = 0

(2m - 1)(m - 1) = 0

⇒ m = 1 or m = 1/2

If m = 1, then the slopes of the lines are 1 and 2.

If m = 1/2, then the slopes of the lines are 1/2 and 1.

Hence, the slopes of the lines are - 1 and - 2 or - 1/2 and - 1 or 1 and 2, or 1/2 and 1

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NCERT Solutions Class 11 Maths Chapter 10 Exercise 10.1 Question 11

The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3, find the slopes of the lines.

Summary:

We found out that the slopes of the lines are - 1 and - 2 or - 1/2 and - 1 or 1 and 2, or 1/2 and 1 given that that tangent of the angles between them is 1/3