The Angle Between Two Intersecting Lines  

The angle between two intersecting lines refers to the measure of the smallest angle formed where the lines cross. This concept plays a crucial role in coordinate geometry, especially in analyzing line orientations, slopes, and their relationships. It is commonly calculated using the slopes of the lines or their direction vectors. In JEE and other competitive exams, questions often involve finding acute angles, checking for perpendicularity, or applying this concept in 3D geometry. Mastery of this topic not only enhances problem-solving speed but also strengthens your understanding of deeper geometric relationships in both 2D and 3D space.

1.0Angle Between Two Intersecting Lines

When two lines intersect, they form two angles that are supplementary. The smaller of these two is generally considered the angle between the lines.

General Idea:

If you know the slopes m_1 and m_2 of the two lines, the angle \theta between them is given by:

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

This gives the acute angle between the lines. Then,

$\theta ={\tan }^{-1}\left(\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|\right)$

Special Cases:

• If $m_1{m}_2=-1,then\theta =90^{\circ }:$ The lines are perpendicular.
• If$m_1=m_2,then\theta =0^{\circ }:$ The lines are parallel.

2.0Finding Slope from Line Equations

If a line is in the form ax + by + c = 0, the slope is:

$m=-\frac{a}{b}$

So, if you're given two general equations, find their slopes first, then use the tangent formula.

3.0Solved Examples for the Angle Between Two Intersecting Lines  

Example 1: Find the angle between the lines:
Line 1: y = 2x + 3
Line 2: y = -x + 5

Solution:

Slopes:

$m_1=2,m_2=-1$

$\tan \theta =\left|\frac{2-(-1)}{1+2\cdot (-1)}\right|=\left|\frac{3}{-1}\right|=3\theta =tan-1(3)\approx 71.56\circ \theta ={\tan }^{-1}(3)\approx 71.56^{\circ }$

Example 2: Find the angle between:
Line 1: 3x + 4y - 7 = 0
Line 2: 5x - 12y + 2 = 0

Solution:

Slopes:

$m_1=-\frac{3}{4},$

$m_2=-\frac{5}{-12}=\frac{5}{12}$

$\tan \theta =\left|\frac{-\frac{3}{4}-\frac{5}{12}}{1+(-\frac{3}{4})\cdot \frac{5}{12}}\right|$

$=\left|\frac{-\frac{9+5}{12}}{1-\frac{15}{48}}\right|$

$=\left|\frac{-\frac{14}{12}}{\frac{33}{48}}\right|$

$=\left|\frac{-\frac{7}{6}}{\frac{33}{48}}\right|$

$=\left|-\frac{56}{33}\right|$

$=\frac{56}{33}$

$\theta ={\tan }^{-1}\left(\frac{56}{33}\right)\approx 59.36^{\circ }$

Example 3: Find the acute angle between the lines y = 2x + 1 and y = 3x - 4.

Solution:

Slopes:

• $m_1=2$
• $m_2=3$

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\left|\frac{2-3}{1+2\cdot 3}\right|=\left|\frac{-1}{7}\right|=\frac{1}{7}$

$\theta ={\tan }^{-1}\left(\frac{1}{7}\right)\approx 8.13^{\circ }$

Answer: $8.13^{\circ }$

Example 4: Find the angle between the lines 4x - 3y + 5 = 0 and 6x + 8y - 7 = 0.

Solution:

Find slopes:

• Line 1: $m_1=-\frac{4}{-3}=\frac{4}{3}$
• Line 2: $m_2=-\frac{6}{8}=-\frac{3}{4}$

$\tan \theta =\left|\frac{\frac{4}{3}+\frac{3}{4}}{1-\frac{4}{3}\cdot \frac{3}{4}}\right|=\left|\frac{\frac{16+9}{12}}{1-1}\right|=\left|\frac{25}{12\cdot 0}\right|$

The denominator is 0, so $\tan \theta \to \infty$

Therefore, the lines are Perpendicular

Example 5: The line y = mx + 1 makes a $45^{\circ }$ angle with the line y = -x + 2. Find the possible values of m.

Solution:

Use formula:

$\tan 45^{\circ }=\left|\frac{m-(-1)}{1+m(-1)}\right|=\left|\frac{m+1}{1-m}\right|$

Since $\tan 45^{\circ }=1:$

$\left|\frac{m+1}{1-m}\right|=1$

Case 1:

$\frac{m+1}{1-m}=1\Rightarrow m+1=1-m\Rightarrow 2m=0\Rightarrow m=0$

Case 2:

$\frac{m+1}{1-m}=-1\Rightarrow m+1=-1+m\Rightarrow 2=0(Notvalid)$

Answer: m = 0

Example 6: Find the angle that the line 3x + 4y = 7 makes with the positive direction of the x-axis.

Solution:

Slope of the line:

$m=-\frac{3}{4}$

Angle with x-axis: $\theta ={\tan }^{-1}\left(\left|m\right|\right)={\tan }^{-1}\left(\frac{3}{4}\right)\approx 36.87^{\circ }$

Answer: $36.87^{\circ }$

Example 7: The line L passes through the point of intersection of x + y = 1 and x - y = 3, and makes an angle of $60^{\circ }$ with the x-axis. Find its equation.

Solution:

Intersection point:

$$\begin{gathered} x+y=1 \\ x-y=3 \\ \Rightarrow 2x=4\Rightarrow x=2,y=-1 \end{gathered}$$

$\text{Slope of line}=\tan 60^{\circ }=\sqrt{3}$

Using point-slope form:

$y+1=\sqrt{3}(x-2)\Rightarrow y=\sqrt{3}x-2\sqrt{3}-1$

Answer: $y=\sqrt{3}x-(2\sqrt{3}+1)$

4.0Practice Questions for the Angle Between Two Intersecting Lines  

1. Find the angle between the lines: $y=\frac{1}{2}x+1andy=-2x+4$
2. Find the acute angle between the lines:
   5x + 12y = 6 and 33x - 4y = 8
3. Find the angle between the x-axis and the line $y=\sqrt{3}x$
4. Find the angle between the lines:
   y - 3x + 2 = 0 and y + x - 1 = 0