Class 11 Maths NCERT Solutions Chapter 9 Straight Lines | Boundless Maths
Chapter 9 Class 11 Maths NCERT Solutions · Unit III · Coordinate Geometry

Class 11 Maths NCERT Solutions Chapter 9: Straight Lines

Free, step-by-step NCERT Solutions for all three exercises of this chapter — the slope of a line and the angle between two lines, every standard form of the equation of a line, and the distance of a point from a line — plus the Miscellaneous Exercise of concurrency, angle bisectors, reflection, and applied geometry problems. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

Straight Lines carries coordinate geometry forward from the distance and section formulas met in Class 10 into a fully algebraic treatment of the line itself. The chapter's central idea is the slope — a single number that captures how steeply a line rises or falls — and everything else follows from it: recognising when two lines are parallel or perpendicular, writing a line's equation in whichever of four equivalent forms best fits the given information, and measuring how far a point sits from a line. The Miscellaneous Exercise then puts all three ideas to work together in richer problems, from proving three lines meet at one point to finding the mirror image of a point in a line.

4Exercises (incl. Misc.)
70Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 9 — Overview

This Class 11 Maths NCERT Solutions Chapter 9 hub covers Straight Lines, the chapter that turns a line — the simplest figure in geometry — into a fully algebraic object. Every non-vertical line has a slope, m = (y2 − y1)/(x2 − x1), which equals tan θ for the line's inclination θ with the positive x-axis. Once the slope is understood, two lines can be checked for parallelism (equal slopes) or perpendicularity (slopes multiplying to −1), and the acute angle between any two intersecting lines can be found from a single formula involving their slopes.

Exercise 9.2 then shows that the same line can be written in several equivalent ways — point-slope form, two-point form, slope-intercept form, and intercept form — each one best suited to a different kind of given information, all reducible to the general form Ax + By + C = 0. Exercise 9.3 adds the distance formulas: the perpendicular distance of a point from a line, and the distance between two parallel lines. The Miscellaneous Exercise brings every idea together in longer applied problems — proving three lines are concurrent, finding the foot of a perpendicular, locating the right bisector of a segment, and reflecting a point in a line as though it were a mirror.

How the Chapter Builds

One Idea Leads to the Next

1

Slope of a Line

Inclination, slope, the angle between two lines, and conditions for parallel/perpendicular lines. Exercise 9.1.

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2

Forms of the Equation of a Line

Point-slope, two-point, slope-intercept, and intercept form, all reducible to Ax + By + C = 0. Exercise 9.2.

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3

Distance of a Point From a Line

The perpendicular distance formula, and the distance between two parallel lines. Exercise 9.3.

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4

Applications

Concurrency, angle bisectors, the foot of a perpendicular, and reflection in a line. Misc. Exercise.

Quick Reference

Important Formulas — Chapter 9

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Straight Lines.

Slope of a Line (§9.2)

Slope from two points

m=\dfrac{y_2-y_1}{x_2-x_1}

Defined whenever x1 ≠ x2 — a vertical line (x1 = x2) has an undefined slope.

Angle between two lines

\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|

Gives the acute angle between two lines with slopes m1 and m2; the obtuse angle is 180° minus this.

Parallel & perpendicular lines

m_1=m_2\qquad m_1m_2=-1

Equal slopes mean the lines are parallel; slopes multiplying to −1 mean the lines are perpendicular.

Forms of the Equation of a Line (§9.3)

Point-slope form

y-y_0=m(x-x_0)

Use when a point on the line and its slope are known.

Slope-intercept & intercept form

y=mx+c\qquad\dfrac{x}{a}+\dfrac{y}{b}=1

Use slope-intercept form when the y-intercept is known, and intercept form when both axis-intercepts are known.

General form

Ax+By+C=0

Every line can be written this way, with A and B not both zero — the form every other equation eventually reduces to.

Distance of a Point From a Line (§9.4)

Distance from a point to a line

d=\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}

The perpendicular distance of the point (x1, y1) from the line Ax + By + C = 0.

Distance between parallel lines

d=\dfrac{|C_1-C_2|}{\sqrt{A^2+B^2}}

Only valid when both lines share the same A and B, i.e. they're genuinely parallel.

Right bisector of a segment

\text{passes through midpoint},\ \perp\text{ to segment}

A recurring Miscellaneous Exercise construction: find the midpoint, then use the negative reciprocal slope.

Decision Guide

Which Straight Lines Technique Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is askingUse thisWhy
Find the slope of a line through two pointsm = (y2 − y1)/(x2 − x1)Direct substitution — remember the slope is undefined for a vertical line (§9.2).
Check whether two lines are parallel or perpendicularCompare slopesEqual slopes (m1 = m2) mean parallel; slopes multiplying to −1 mean perpendicular (§9.2).
Write the equation of a line from given informationPick the matching formPoint + slope → point-slope; two points → two-point; intercepts → intercept form (§9.3).
Find how far a point is from a linePerpendicular distance formulad = |Ax1 + By1 + C| / √(A² + B²), using the line's general form (§9.4).
Find the distance between two parallel linesParallel-line distance formulaOnly works once both equations share identical A and B coefficients (§9.4).
Show three lines all pass through one pointSolve two, check the thirdFind the intersection of any two lines, then verify it also satisfies the third equation (Misc. Exercise).
Find the mirror image of a point in a linePerpendicular bisector constructionThe line is the perpendicular bisector of the segment joining the point and its image (Misc. Exercise).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all three exercises.

  • Swapping the order in the slope formula — m = (y2 − y1)/(x2 − x1) needs the same point subtracted first in both numerator and denominator; mixing the order flips the sign of the slope.
  • Forgetting the absolute value in the distance formula — distance is always non-negative, so the modulus around Ax1 + By1 + C must never be dropped, even when the expression inside is negative.
  • Applying the perpendicular-line distance formula to non-parallel lines — the formula d = |C1 − C2|/√(A²+B²) only works when both equations already share the same A and B; check this before using it.
  • Picking the wrong angle — the standard formula gives the acute angle between two lines; if the question asks for the obtuse angle instead, it must be found as 180° minus the acute one.
  • Sign errors converting between forms — moving between intercept form and general form, or slope-intercept and general form, is a common place to drop or flip a sign partway through.
  • Not checking both sign cases when squaring — problems that involve squaring an equation (perpendicularity conditions, equal-intercept problems) often have two valid solutions; dropping one because only the "obvious" sign was checked is a frequent slip in the Miscellaneous Exercise.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 9 — Choose an Exercise

9.1

Exercise 9.1

Slope of a Line — inclination, slope, angle between two lines, parallel and perpendicular conditions · 11 questions

Solve Exercise 9.1 →
9.2

Exercise 9.2

Various Forms of the Equation of a Line — point-slope, two-point, slope-intercept, and intercept form · 19 questions

Solve Exercise 9.2 →
9.3

Exercise 9.3

Distance of a Point From a Line — perpendicular distance, distance between parallel lines, right bisectors · 17 questions

Solve Exercise 9.3 →
M

Miscellaneous Exercise

Concurrency, angle bisectors, reflection in a line, and applied straight-line geometry · 23 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Straight Lines — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →
Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 9, Straight Lines.

How many exercises are there in Chapter 9, Straight Lines?
There are three main exercises — 9.1 (Slope of a Line, 11 questions), 9.2 (Various Forms of the Equation of a Line, 19 questions), and 9.3 (Distance of a Point From a Line, 17 questions) — plus a Miscellaneous Exercise of 23 questions covering mixed applications, totalling 70 questions.
What is the slope of a line?
The slope (or gradient) of a line is m = tan θ, where θ is the inclination the line makes with the positive direction of the x-axis, measured anticlockwise. If two points (x1, y1) and (x2, y2) lie on the line, the slope is m = (y2 − y1)/(x2 − x1). The slope of a horizontal line is zero, and the slope of a vertical line is undefined.
What are the different forms of the equation of a line?
A line can be written in point-slope form y − y0 = m(x − x0), two-point form using two known points, slope-intercept form y = mx + c, or intercept form x/a + y/b = 1. Any equation of the form Ax + By + C = 0, with A and B not both zero, is called the general linear equation of a line.
Where can I find the official NCERT textbook for this chapter?
Straight Lines is Chapter 9 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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