Find the angles between each of the following pairs of straight lines:

1.	Find the angles between each of the following pairs of straight lines:(i) 3x + y + 12 = 0 and x + 2y – 1 = 0(ii) 3x – y + 5 = 0 and x – 3y + 1 = 0
Answer» (i) 3x + y + 12 = 0 and x + 2y – 1 = 0 Given: The equations of the lines are 3x + y + 12 = 0 … (1) x + 2y − 1 = 0 … (2) Let m₁ and m₂ be the slopes of these lines. m₁ = -3, m₂ = -1/2 Let θ be the angle between the lines. Then, by using the formula tan θ = [(m₁ – m₂) / (1 + m₁m₂)] = [(-3 + 1/2) / (1 + 3/2)] = 1 So, θ = π/4 or 45^o ∴ The acute angle between the lines is 45° (ii) 3x – y + 5 = 0 and x – 3y + 1 = 0 Given: The equations of the lines are 3x − y + 5 = 0 … (1) x − 3y + 1 = 0 … (2) Let m₁ and m₂ be the slopes of these lines. m₁ = 3, m₂ = 1/3 Let θ be the angle between the lines. Then, by using the formula tan θ = [(m₁ – m₂) / (1 + m₁m₂)] = [(3 + 1/3) / (1 + 1)] = 4/3 So, θ = tan^-1 (4/3) ∴ The acute angle between the lines is tan^-1 (4/3).

Find the angles between each of the following pairs of straight lines:(i) 3x + y + 12 = 0 and x + 2y – 1 = 0(ii) 3x – y + 5 = 0 and x – 3y + 1 = 0

Answer»

(i) 3x + y + 12 = 0 and x + 2y – 1 = 0

Given:

The equations of the lines are

3x + y + 12 = 0 … (1)

x + 2y − 1 = 0 … (2)

Let m₁ and m₂ be the slopes of these lines.

m₁ = -3, m₂ = -1/2

Let θ be the angle between the lines.
Then, by using the formula

tan θ = [(m₁ – m₂) / (1 + m₁m₂)]

= [(-3 + 1/2) / (1 + 3/2)]

= 1

So,

θ = π/4 or 45^o

∴ The acute angle between the lines is 45°

(ii) 3x – y + 5 = 0 and x – 3y + 1 = 0

Given:

The equations of the lines are

3x − y + 5 = 0 … (1)

x − 3y + 1 = 0 … (2)

Let m₁ and m₂ be the slopes of these lines.

m₁ = 3, m₂ = 1/3

Let θ be the angle between the lines.
Then, by using the formula

tan θ = [(m₁ – m₂) / (1 + m₁m₂)]

= [(3 + 1/3) / (1 + 1)]

= 4/3

So,

θ = tan^-1 (4/3)

∴ The acute angle between the lines is tan^-1 (4/3).