JEE Maths — Straight Lines and Pair of Lines Complete Chapter Guide

Chapter Overview & Weightage

Straight Lines is one of those chapters that rewards systematic preparation. Every year, JEE Main guarantees 1-2 questions from this chapter, and JEE Advanced has used it to test deeper reasoning through pair of lines and family of lines.

Weightage snapshot: 4-5% in JEE Main (roughly 1-2 questions per session). JEE Advanced uses this chapter in tandem with Circles and Conics — a standalone Straight Lines question appeared in JEE Advanced 2023 Paper 1 involving the foot of perpendicular and a parametric locus.

Year	JEE Main (Questions)	Marks	Topic Focus
2024	2	8	Distance formula, family of lines
2023	1	4	Angle bisectors
2022	2	8	Pair of straight lines, slope forms
2021	1	4	Concurrence condition
2020	2	8	Foot of perpendicular, reflection

The pattern is clear: distance-based problems and pair of lines dominate. Slope forms and standard line equations are setup tools — the actual question usually lives in one of those two zones.

Key Concepts You Must Know

Prioritised by how often they appear in PYQs:

Tier 1 — Must be exam-ready:

Slope of a line, collinearity, and the angle between two lines
Distance of a point from a line and between two parallel lines
Family of lines passing through the intersection of two given lines: $L_1 + \lambda L_2 = 0$
Foot of perpendicular and reflection of a point across a line
Condition for three lines to be concurrent

Tier 2 — High-value but less frequent:

Angle bisectors of two lines (and which bisector is acute/obtuse)
Pair of straight lines: combined equation $ax^2 + 2hxy + by^2 = 0$
Angle between pair of lines, conditions for perpendicularity and coincidence
Homogenisation of a curve with a line

Tier 3 — Know the concept, not the derivation:

Shifting of origin and rotation of axes
Normal form of a line

Important Formulas

Slope: $m = \tan\theta = \dfrac{y_2 - y_1}{x_2 - x_1}$

Angle between two lines with slopes $m_1$ , $m_2$ :

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

When to use: Any problem asking “find the angle” or “find the acute angle bisector direction.” Also use this to check perpendicularity ( $m_1 m_2 = -1$ ) or parallelism ( $m_1 = m_2$ ).

Distance of point $(x_1, y_1)$ from line $ax + by + c = 0$ :

d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}

Distance between parallel lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$ :

d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}

When to use: Anywhere “distance” appears — from a point to a line, between parallel lines, or to find foot of perpendicular. Also the base formula for area of a triangle using vertex-to-base distance.

If $L_1 \equiv a_1x + b_1y + c_1 = 0$ and $L_2 \equiv a_2x + b_2y + c_2 = 0$ , then any line through their intersection is:

L_1 + \lambda L_2 = 0

When to use: When a problem says “a line through the intersection of $L_1$ and $L_2$ satisfies condition X — find it.” Write $L_1 + \lambda L_2 = 0$ , apply the condition, solve for $\lambda$ . This avoids solving two equations to find the intersection point first.

Foot of perpendicular from $(x_1, y_1)$ to $ax + by + c = 0$ :

\frac{x - x_1}{a} = \frac{y - y_1}{b} = -\frac{ax_1 + by_1 + c}{a^2 + b^2}

Reflection $(x', y')$ of $(x_1, y_1)$ across the line: the foot is the midpoint of $(x_1, y_1)$ and $(x', y')$ .

When to use: Locus problems involving images/reflections, and any question where “foot of perpendicular” is explicitly asked. In JEE Advanced, these appear wrapped inside a larger locus problem.

For $ax^2 + 2hxy + by^2 = 0$ (homogeneous, passes through origin):

Angle between the pair:

\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}

Perpendicular pair: $a + b = 0$

Coincident lines: $h^2 = ab$

For $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ (general pair):

\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0

When to use: Whenever a second-degree equation in $x$ and $y$ appears — check if $\Delta = 0$ to confirm it’s a pair of lines. Then use the above angle formula.

Solved Previous Year Questions

PYQ 1 — JEE Main 2024 Shift 1

Q. If the distance between the lines $3x - 4y + 7 = 0$ and $3x - 4y + k = 0$ is 3, find the sum of possible values of $k$ .

Solution:

Two parallel lines $3x - 4y + 7 = 0$ and $3x - 4y + k = 0$ have the same normal direction, so the distance formula applies directly.

d = \frac{|k - 7|}{\sqrt{9 + 16}} = \frac{|k - 7|}{5} = 3

|k - 7| = 15

k - 7 = 15 \implies k = 22 \quad \text{or} \quad k - 7 = -15 \implies k = -8

Sum of possible values $= 22 + (-8) = \boxed{14}$

The question asked for “sum of possible values” — a signal that there are two valid answers. Always check both cases of the modulus.

PYQ 2 — JEE Main 2023 January Session

Q. A line passing through the intersection of $x + 2y - 3 = 0$ and $2x - y + 1 = 0$ is perpendicular to $3x + 4y = 0$ . Find its equation.

Solution:

Any line through the intersection: $(x + 2y - 3) + \lambda(2x - y + 1) = 0$

x(1 + 2\lambda) + y(2 - \lambda) + (-3 + \lambda) = 0

Slope of this line $= -\dfrac{1 + 2\lambda}{2 - \lambda}$

Slope of $3x + 4y = 0$ is $-\dfrac{3}{4}$ .

For perpendicularity, product of slopes $= -1$ :

\left(-\frac{1 + 2\lambda}{2 - \lambda}\right) \times \left(-\frac{3}{4}\right) = -1

\frac{3(1 + 2\lambda)}{4(2 - \lambda)} = -1

3 + 6\lambda = -8 + 4\lambda

2\lambda = -11 \implies \lambda = -\frac{11}{2}

Substituting $\lambda = -\dfrac{11}{2}$ :

1 + 2\left(-\frac{11}{2}\right) = 1 - 11 = -10

2 - \left(-\frac{11}{2}\right) = 2 + \frac{11}{2} = \frac{15}{2}

-3 + \left(-\frac{11}{2}\right) = -\frac{17}{2}

Line: $-10x + \dfrac{15}{2}y - \dfrac{17}{2} = 0 \implies -20x + 15y - 17 = 0 \implies 20x - 15y + 17 = 0$

\boxed{20x - 15y + 17 = 0}

PYQ 3 — JEE Main 2022 July Session

Q. The combined equation of the lines $y = x$ and $y = -x$ is:

This is a concept-application question testing pair of lines.

Solution:

$y = x$ means $y - x = 0$ and $y = -x$ means $y + x = 0$ .

Combined equation: $(y - x)(y + x) = 0 \implies y^2 - x^2 = 0 \implies x^2 - y^2 = 0$

Now checking the standard form: $ax^2 + 2hxy + by^2 = 0$ with $a = 1$ , $h = 0$ , $b = -1$ .

Angle: $\tan\theta = \dfrac{2\sqrt{0 - (1)(-1)}}{1 + (-1)} = \dfrac{2}{0}$ — undefined, so $\theta = 90°$ . The two lines are perpendicular, which checks out since $a + b = 1 - 1 = 0$ .

\boxed{x^2 - y^2 = 0}

Difficulty Distribution

Difficulty	% of Questions	What to Expect
Easy	40%	Standard distance formula, slope calculations, equation of line in given form
Medium	45%	Family of lines, foot of perpendicular, concurrent lines, angle bisectors
Hard	15%	Pair of straight lines with homogenisation, locus problems, JEE Advanced type

In JEE Main, 1 question is almost always medium-difficulty involving either family of lines or foot of perpendicular. The second question (when it appears) is usually easy — a direct formula application. Don’t over-prepare for the hard end in JEE Main.

Expert Strategy

How toppers actually prepare this chapter:

Start with slope and line forms — but spend minimal time here. You need these as tools, not as exam content. The actual marks live in distance, family of lines, and pair of lines.

Week 1 priority: Master foot of perpendicular and reflection cold. Write the formula 10 times and do 5 problems from each of Cengage or Arihant. These two concepts together cover nearly 30% of the PYQs in this chapter.

Week 2 priority: Pair of straight lines. Learn the $\Delta = 0$ condition for recognising a pair. Practice identifying $a$ , $h$ , $b$ , $g$ , $f$ , $c$ from a given equation — students lose time here in exams.

For family of lines problems, always write $L_1 + \lambda L_2 = 0$ first — never find the intersection point manually unless forced. It saves 90 seconds per problem and avoids arithmetic errors.

For JEE Advanced: Connect Straight Lines with Circles. A favourite Advanced problem type: “Find the chord of contact / equation of chord” — this mixes both chapters. Practice homogenisation (joining the equation of a pair of lines to a conic using a given line).

PYQ drill order: Solve the last 5 years of JEE Main PYQs by chapter. For this chapter, 20-25 PYQs will cover every formula and concept you need. Pattern recognition matters more than grinding new problems.

Common Traps

Trap 1 — The sign in the distance formula. Students write $\dfrac{ax_1 + by_1 + c}{\sqrt{a^2+b^2}}$ without the absolute value bars and get a negative distance. Distance is always non-negative. The modulus is not optional.

Trap 2 — Slope of a perpendicular line. If $m$ is the slope of a line, the perpendicular slope is $-1/m$ , not $-m$ . In exam pressure, students write $-m$ . Double-check this in any problem involving “perpendicular from a point.”

Trap 3 — Confusing the two angle bisectors. Given two lines, there are always two angle bisectors — one acute, one obtuse. Questions often ask for the “acute angle bisector” or “bisector of the angle containing the origin.” Use the sign of $\dfrac{ax_1 + by_1 + c_1}{\sqrt{a_1^2+b_1^2}}$ and $\dfrac{ax_1 + by_1 + c_2}{\sqrt{a_2^2+b_2^2}}$ (with the origin as the test point) to decide which is which.

Trap 4 — Checking if a second-degree equation is actually a pair of lines. Before applying pair-of-lines formulas, verify $\Delta = 0$ . Examiners have set questions where a second-degree equation is a circle or conic — not a pair — and students blindly apply the angle formula and get a wrong answer.

Trap 5 — The family of lines parameter. $L_1 + \lambda L_2 = 0$ passes through the intersection of $L_1 = 0$ and $L_2 = 0$ , but it cannot represent $L_2 = 0$ itself (no value of $\lambda$ gives that). If the answer requires $L_2$ as a candidate, check it separately.