Careers360 Logo
NIT Cutoff 2025 for B.Tech Metallurgical and Materials Engineering

Angle Between Two Lines - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Angle between two straight line is considered one of the most asked concept.

  • 54 Questions around this concept.

Solve by difficulty

The angle between the lines represented by \mathrm{x^{2}-7 x y+12 y^{2}=0} is:

The portion of the line $4 x+5 y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ is :

Concepts Covered - 1

Angle between two straight line

Angle between two straight line

Two lines are given with the slope mand m2, then acute angle θ between the lines is given by 

\theta=\tan ^{-1}\left|\frac{\mathrm{m}_{1}-\mathrm{m}_{2}}{1+\mathrm{m}_{1} \mathrm{m}_{2}}\right|.
 

\\\mathrm{Let \,m_{1} \;and \;\mathrm{m}_{2}\; be\; the \;slope\; of\; two\; given\; straight\; lines \;and\; \theta_{1}\; and \;\theta_{2}\; is\; the\; inclinations}\\\therefore \mathrm{m}_{1}=\tan \theta_{1} \text { and } \mathrm{m}_{2}=\tan \theta_{2}\\ \text{let } \theta \text{ and } \pi-\theta$ be the angles between straight line $\left(\theta \neq \frac{\pi}{2}\right)\\\text{from the figure}\\ \theta_{2}=\theta+\theta_{1} \quad \text { or } \quad \theta=\theta_{1}-\theta_{2} \\ \tan (\theta)=\tan \left(\theta_{1}-\theta_{2}\right) \\ \tan (\theta)=\left(\frac{\tan \left(\theta_{2}\right)-\tan \theta_{1}}{1+\tan \left(\theta_{1}\right) \tan \left(\theta_{2}\right)}\right)=\left(\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{2} \mathrm{m}_{1}}\right)\\\text { Also, } \tan (\pi-\theta)=-\tan (\theta)=-\left(\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{2} \mathrm{m}_{1}}\right)

\\\Rightarrow \theta=\tan ^{-1}\left[\pm\left(\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{2} \mathrm{m}_{1}}\right)\right]\\ \text{Hence, the acute angle between two straight lines is given as} \\\mathbf{\theta=\tan ^{-1}\left|\left(\frac{m_{2}-m_{1}}{1+m_{2} m_{1}}\right)\right|}

 

Note:

1. If the angle between the two lines is 0or \pi then lines are parallel two each other.  In this case, m1 = m2 where m1 and m2 are slopes of two lines.

2. If the angle between  the two lines is \frac{\pi}{2} or -\frac{\pi}{2} then lines are perpendicular two each other. Then in this case m1⋅m2 = -1 where, m1 and m2 are slopes of two lines.

3. Equation of two straight line given as \mathrm{A}_{1} \mathrm{x}+\mathrm{B}_{1} \mathrm{y}+\mathrm{C}_{1}=0 and \mathrm{A}_{2} \mathrm{x}+\mathrm{B}_{2} \mathrm{y}+\mathrm{C}_{2}=0. If these two lines are coincident then,

\mathrm{\frac{A_{1}}{A_{2}}=\frac{B_{1}}{B_{2}}=\frac{C_{1}}{C_{2}}}

 

Illustriation

Find the angle between the line joining the points (0, 0), (2, 6) with line joining the points (2, 3), (3, 4)

Let A = (0, 0), B = (2, 6), C = (2, 3) and D = (3, 4)

Let m1 is the slope of AB and m2 is slope of CD

\\\mathrm{m_1=\frac{6-0}{2-0}=3\;\;and\;\;m_2=\frac{4-3}{3-2}=1}\\\mathrm{Let\;\theta\;be\;the\;acute\;angle\;between\;the\;lines\;AB\;and\;CD}\\\mathrm{\tan\theta=\left | \frac{m_1-m_2}{1+m_1m_2} \right |=\left | \frac{3-1}{1+3\times1} \right |=\frac{1}{2}}\\\mathrm{\theta=\tan^{-1}\left ( \frac{1}{2} \right )}

Study it with Videos

Angle between two straight line

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Books

Reference Books

Angle between two straight line

Mathematics for Joint Entrance Examination JEE (Advanced) : Coordinate Geometry

Page No. : 1.17

Line : 10

E-books & Sample Papers

Get Answer to all your questions

Back to top