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

(i) 

   

 According to ques,:

The equations of the lines are

   

   

Let m₁ and m₂ be the slopes of these lines.

   

Let θ be the angle between the lines.

Then, by using the formula

   

   

Or

   

So,

   

∴ The acute angle between the lines is 45°

   

According to ques,:

The equations of the lines are

   

   

Let m₁ and m₂ be the slopes of these lines.

   

Let θ be the angle between the lines.

Then, by using the formula

   

   

Or

   

So,

   

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