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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

EASY

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

$45^{\circ}$

$60^{\circ}$

$15^{\circ}$

None of these

View Solution

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 :

$\frac{8}{5}$

$\frac{25}{41}$

$\frac{2}{5}$

$\frac{30}{41}$

View Solution

Concepts Covered - 1

Angle between two straight line

Angle between two straight line

Two lines are given with the slope m₁and 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 0^oor $\pi$ 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

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