Use tan θ = |m₁ - m₂| / (1 + m₁m₂).

Angle Between Two Lines m₁ and m₂

Question

Two lines have slopes $m_1 = 2$ and $m_2 = \frac{1}{3}$ . Find the acute angle between them.

Solution — Step by Step

The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by:

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

The absolute value ensures we always get the acute angle (the smaller of the two angles formed).

Plug in $m_1 = 2$ and $m_2 = \frac{1}{3}$ :

\tan\theta = \left|\frac{2 - \frac{1}{3}}{1 + 2 \cdot \frac{1}{3}}\right|

Work the numerator and denominator separately — this is where most mistakes happen.

Numerator: $2 - \frac{1}{3} = \frac{6-1}{3} = \frac{5}{3}$

Denominator: $1 + \frac{2}{3} = \frac{3+2}{3} = \frac{5}{3}$

\tan\theta = \left|\frac{\frac{5}{3}}{\frac{5}{3}}\right| = |1| = 1

\tan\theta = 1 \implies \theta = 45°

The acute angle between the two lines is 45°.

Why This Works

When two lines meet, they form two pairs of vertically opposite angles. One pair is acute (or right), the other is obtuse — they add up to 180°. We always want the acute angle, which is why the formula uses the absolute value.

The formula itself comes from the tangent subtraction identity: $\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}$ . Since slope = $\tan$ of inclination angle, the difference of inclination angles gives us the angle between the lines.

Notice what happens when $1 + m_1 m_2 = 0$ : the denominator blows up, meaning $\tan\theta \to \infty$ , so $\theta = 90°$ . That’s why the perpendicularity condition is $m_1 m_2 = -1$ .

Alternative Method

We can verify using direction vectors. Line 1 with slope 2 has direction vector $\vec{a} = (1, 2)$ . Line 2 with slope $\frac{1}{3}$ has direction vector $\vec{b} = (3, 1)$ .

\cos\theta = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}||\vec{b}|} = \frac{|1\cdot3 + 2\cdot1|}{\sqrt{5} \cdot \sqrt{10}} = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}

This gives $\theta = 45°$ — same answer. This dot product method is useful when lines are given in vector form, which appears in 3D geometry (a Class 12 extension of this same idea).

Common Mistake

Students often forget the absolute value and compute $\tan\theta$ as a negative number, then get confused when $\theta$ comes out negative or obtuse. The formula gives the tangent of the acute angle — always take the absolute value of the entire expression before finding the inverse tan. If $\tan\theta < 0$ after applying the formula (without mod), it means the angle you computed is obtuse; the acute angle is its supplement.

In JEE Main 2023, this formula appeared in a coordinate geometry question where the angle was given and you had to find an unknown slope — essentially working the formula backwards. Set up $\tan\theta = k$ and solve the resulting equation in $m$ . You’ll get a quadratic, giving two possible slopes (two lines at the same angle to the given line).

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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