NDA Maths · Lines

Angle Between Lines, Parallel & Perpendicular

The angle between two lines from their slopes, and the slope conditions for lines to be parallel or perpendicular.

Why this matters

The angle formula and the parallel/perpendicular tests are short, high-frequency tools — used directly and inside triangle and quadrilateral problems. The only trap is the sign in the tangent formula (acute vs obtuse).

Concept 1 of 2

Angle between two lines

Intuition

The angle between two lines depends only on their slopes. The tangent formula gives the acute angle when you take the absolute value; drop the bars (or take the supplement) for the obtuse one.

Definition

For slopes $m_1,m_2$ : $\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1 m_2}\right|$ gives the acute angle; the obtuse angle is its supplement. If $1+m_1m_2=0$ the lines are perpendicular ( $\theta=90°$ ). For lines given as $a_1x+b_1y+c_1=0$ , use slopes $-a_i/b_i$ .

Worked example

Find the acute angle between lines of slopes

1

and

\tfrac13

$\tan\theta=\left|\dfrac{1-\tfrac13}{1+1\cdot\tfrac13}\right|=\dfrac{2/3}{4/3}=\tfrac12$ .
$\theta=\tan^{-1}\tfrac12$ .

Answer:

\tan^{-1}\tfrac12

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the obtuse angle between lines with slopes

2-\sqrt3

and

2+\sqrt3

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Angle formula between slopes $m_1,m_2$ ?
2.
Absolute value gives which angle?
3.
$1+m_1m_2=0$ means?
4.
Obtuse angle relates to acute how?

From the bank · past-year question

Example 1LinesMODERATE

What is the obtuse angle between the lines whose slopes are

2-\sqrt{3}

and

2+\sqrt{3}

[Q59 · Apr · 2020]

Drill 11 more on angle between two lines

Concept 2 of 2

Parallel and perpendicular conditions

Intuition

Two lines are parallel when their slopes match, and perpendicular when the slopes multiply to

-1

. In coefficient form these become clean conditions on

a,b

Definition

Parallel: $m_1=m_2$ ; for $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ , parallel iff $a_1b_2=a_2b_1$ (i.e. $\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}$ ). Perpendicular: $m_1 m_2=-1$ , i.e. $a_1a_2+b_1b_2=0$ .

Worked example

Are

2x+3y=5

and

3x-2y=7

perpendicular?

$a_1a_2+b_1b_2=2(3)+3(-2)=6-6=0$ .
The condition holds.

Answer:Yes — perpendicular.

Practice this conceptself-check · 4 quick reps

Try it yourself

Under what condition are

ax+by+c=0

and

a'x+b'y+c'=0

parallel?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Parallel condition on slopes?
2.
Perpendicular condition on slopes?
3.
Perpendicular in coefficients?
4.
Are $2x+3y=5$ , $3x-2y=7$ perpendicular?

From the bank · past-year question

Example 2LinesMODERATE

Under which one of the following conditions are the lines

ax+by+c=0

and

bx+ay+c=0

parallel (

a\neq0, b\neq0

[Q88 · Apr · 2022]

Drill 3 more on parallel and perpendicular conditions

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1LinesMODERATE

Consider the following statements: Statement I: If the line segment joining the points P(m, n) and Q(r, s) subtends an angle

\alpha

at the origin, then

\cos\alpha=\frac{ms-nr}{\sqrt{(m^2+n^2)(r^2+s^2)}}

. Statement II: In any triangle ABC, it is true that

a^2=b^2+c^2-2bc\cos A

. Which one of the following is correct in respect of the above two statements?

[Q60 · Sep · 2018]

Example 2LinesEASY

The points

(-a,-b)

(0,0)

(a,b)

and

(a^2,ab)

are:

[Q82 · Sep · 2023]

Example 3LinesMODERATE

What is the acute angle between the pair of straight lines

\sqrt{2}x+\sqrt{3}y=1

and

\sqrt{3}x+\sqrt{2}y=2

[Q59 · Apr · 2017]

Example 4LinesEASY

If the lines

y + px = 1

and

y - qx = 2

are perpendicular, then which one of the following is correct?

[Q58 · Apr · 2021]

Example 5LinesMODERATE

The angle between the two lines

lx + my + n = 0

and

l'x + m'y + n' = 0

is given by

\tan^{-1}\theta

. What is

\theta

equal to?

[Q53 · Sep · 2018]

Drill every past-year question on this subtopic

16 questions from the bank — paginated, with cart and Word-export support.

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