Test: Straight Lines: Angle between Lines (6 July) - JEE MCQ

Concept: 
The slope of a line of the form y = mx + c is 
m = tan θ and θ = tan^-1(m)
The angle between the two lines with slopes m₁ and m₂ is,

Calculation: 
Given, the slope of a line L is 2.
The angle between the two lines with slopes m₁ and m₂ is,

Given that the lines with slopes m₁ and m₂ are inclined at an angle (π/6) with L.

Given:
The lines which make an angle of 45° with the line 3x - y + 4 = 0 
Concept:
Angle θ between two lines having slopes m₁ and m₂ is given by the relation tanθ = 
Solution:
Slope of the line 3x - y + 4 = 0 is 3
Let slope of other line be m
Then, using above concept -

Solving, we get m = -2 and m = 1/2

Concept:
Equation of two straight lines which pass through a point (x₁, y₁) and make a given angle α with the given straight line y = mx + c, are :

Calculation:
Here, x₁ = 1, y₁ = 1, α = 60° and m = -1(slope of line x + y - 3 = 0)
∴ The equation of lines are:

⇒ ( 1 - √3) y - ( 1 - √3) =  ( - 1 - √3) x + (1 + √3)
⇒  ( 1 - √3) y - 1 + √3 = - (1 + √3) x + 1 + √3
⇒ (1 + √3) x + ( 1 - √3) y - 1 + √3 - 1 -  √3 = 0
⇒ (1 + √3) x + ( 1 - √3) y  - 2 = 0

∴ Required equation of line is (1 + √3)x + (1 - √3) y - 2 = 0.

Concept:

• The angle θ between the two lines y = m₁x + c₁ and y = m₂x + c₂, is given by:
  
• The slope (m) of the line passing through the points (x₁, y₁) and (x₂, y₂) is given by:
  

Calculation:
Given that A = (1, 3), B = (2, 2) and C = (3, 4).
The angle B in the triangle ABC is the angle between the lines BA and BC.

Angle B = 

CONCEPT:
If θ is the acute angle between two non-vertical and non-perpendicular lines L₁ and L₂ with slopes m₁ and m₂ respectively then 
CALCULATION:
Here, we have to find the angle between the lines whose slopes are 1/2 and 3
Let m₁ = 1/2 and m₂ = 3
As we know that, 

So, the angle between the lines whose slopes are 1/2 and 3 is 45° 
Hence, option B is the correct answer.

Concept:
The angle between the lines y = m₁x + c₁ and y = m₂x + c₂ is given by  tan θ = 
Calculation:
Given lines are y = x + 4 and y =  2x - 3
Let slope of 1st and 2nd line are m₁ and m₂ respectively,
Therefore, m₁ = 1 and m₂ = 2
As we know, tan θ =  

Concept:
The angle between the lines y = m₁x + c₁ and y = m₂x + c₂ is given by  tan θ =  
Calculation:
Given lines are  y = 
Let slope of 1st and 2nd line are m₁ and m₂ respectively,

CONCEPT:
If α is the acute angle between two non-vertical and non-perpendicular lines L₁ and L₂ with slopes m₁ and m₂ respectively then 
CALCULATION:
Here, we have to find the angle between the lines whose slopes are 


So, the angle between the lines whose slopes are 
Hence, option C is the correct answer.

The angle between two lines 
If θ is the angle between two intersecting lines defined by y = m₁x + c₁ and y = m₂x + c₂, then, the angle θ is given by

Calculation:
Given lines are
2x - y = 3      ....(1)
and x - 2y = 3      ...(2)
From equation (1),
2x - y = 3 
⇒ y = 2x - 3
Here, m₁ = 2
From equation (2),
x - 2y = 3 
2y = x - 3

The angle between the lines 2x - y = 3 and x - 2y = 3 is tan^-1(3/4)

Concept:
Angle between two lines: The angle θ between the lines having slope m₁ and m₂ is given by

Calculation:
Given: y - √3x – 5 = 0 & √3y – x + 6 = 0
y - √3x – 5 = 0
⇒ y = √3x + 5
So, slope of line, m₁ = √3
√3y – x + 6 = 0

So, slope of the line, m₂ = 
Let θ be the acute angle between the lines.

⇒ θ = 30°