Straight Lines Class 11 Notes & Important Questions (Ch 9)

Revision notes

Straight Lines: revision notes

1. Slope

The slope of a non-vertical line making angle $θ$ with the positive $x$ -axis is $m = \tan θ$ . Through $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ : $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Horizontal lines have slope $0$ ; vertical lines have no slope.

Parallel lines: $m_{1} = m_{2}$ . Perpendicular lines: $m_{1} m_{2} = - 1$ .
Three points are collinear if the slope between the first two equals the slope between the last two.

2. Angle between two lines

If $θ$ is the acute angle between lines of slopes $m_{1}, m_{2}$ ( $1 + m_{1} m_{2} \neq 0$ ): $\tan θ = | \frac{m_{2} - m_{1}}{1 + m_{1} m_{2}} |$ .

3. Forms of the equation of a line

Parallel to the axes: $x = a$ , $y = b$ .
Point-slope: $y - y_{1} = m (x - x_{1})$ . Slope-intercept: $y = m x + c$ .
Two-point: $y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$ .
Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ (intercepts $a$ and $b$ ).
General form $A x + B y + C = 0$ : slope $- \frac{A}{B}$ , $x$ -intercept $- \frac{C}{A}$ , $y$ -intercept $- \frac{C}{B}$ .

4. Distances

Distance of $(x_{1}, y_{1})$ from $A x + B y + C = 0$ : $d = \frac{| A x_{1} + B y_{1} + C |}{\sqrt{A^{2} + B^{2}}}$ . Distance between the parallel lines $A x + B y + C_{1} = 0$ and $A x + B y + C_{2} = 0$ : $\frac{| C_{1} - C_{2} |}{\sqrt{A^{2} + B^{2}}}$ (make the $x$ and $y$ coefficients identical first).

Worked example 1

Find the equation of the line through $(2, - 1)$ parallel to $3 x - y + 4 = 0$ .

Slope $3$ : $y + 1 = 3 (x - 2)$ , i.e. $3 x - y - 7 = 0$ .

Worked example 2

Find the distance of $(1, 2)$ from $4 x + 3 y - 20 = 0$ .

$\frac{| 4 + 6 - 20 |}{5} = 2$ .

Worked example 3

Find the acute angle between $y = \sqrt{3} x$ and the $x$ -axis.

$\tan θ = \sqrt{3}$ , so $θ = 60^{\circ}$ .

Common errors

Slope written as $\frac{x_{2} - x_{1}}{y_{2} - y_{1}}$ , or with the points in a different order on top and bottom.
Forgetting the modulus in the angle and distance formulae.
Distance between parallel lines without first making the coefficients of $x$ and $y$ the same.
Slope of $A x + B y + C = 0$ taken as $\frac{A}{B}$ instead of $- \frac{A}{B}$ .

Exam tips

Give final equations in the general form $A x + B y + C = 0$ with integer coefficients.
For perpendicular lines, write the new slope as $- \frac{1}{m}$ before substituting.

Topics in this chapter: Slope · Forms of the equation of a line · Distances · Angle between two lines.

Formulas for this chapter: Chapter 9 on the Class 11 formula sheet (free PDF and A3 poster).

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can find slopes, test parallel and perpendicular lines and write simple line equations.

Read first: 1. Slope; 3. Forms of the equation of a line 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can use every form of the line equation, test collinearity and find distances from a point to a line.

Read first: 2. Angle between two lines; 3. Forms; 4. Distances; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can combine line equations, distances and areas in triangle problems.

Read first: 3. Two-point form; 4. Distance of a point from a line 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can find lines at a given angle, reflections of points and distances between parallel lines.

Read first: 2. Angle between lines; 4. Parallel lines; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 32 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceSlope

The slope of the line through $(2, 3)$ and $(6, 11)$ is

(a) $\frac{1}{2}$
(b) $2$
(c) $8$
(d) $- 2$

Q2·1 mark·Multiple choiceSlope

The slope of a line inclined at $135^{\circ}$ to the positive $x$ -axis is

(a) $- 1$
(b) $1$
(c) $\sqrt{3}$
(d) $- \frac{1}{\sqrt{3}}$

Q3·1 mark·Multiple choiceForms of the equation of a line

The line through $(1, - 2)$ with slope $3$ is

(a) $x - 3 y - 7 = 0$
(b) $3 x - y + 5 = 0$
(c) $3 x - y - 5 = 0$
(d) $3 x + y - 1 = 0$

Worked solution·2 marks·Very short answerAngle between two lines

Find the acute angle between the lines $y = x$ and $y = (2 + \sqrt{3}) x$ .

Full worked solution Free, no sign-in

$\tan θ$ $= | \frac{(2 + \sqrt{3}) - 1}{1 + (2 + \sqrt{3})} |$ $= \frac{1 + \sqrt{3}}{3 + \sqrt{3}} = \frac{1}{\sqrt{3}}$ M1
$θ = 30^{\circ}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S25 · Problem S89 · Problem S24 · Problem S88. All 22 →

Knowledge organiser: Straight Lines on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Equation of a straight line. All formulas A–Z

Where marks are lost in Straight Lines

Slope of Ax + By + C = 0 taken as A/B. Fix: rearrange to y = −(A/B)x − C/B.
Modulus dropped in the distance formula, giving a negative distance. Fix: |Ax₁ + By₁ + C|/√(A² + B²).
Parallel-line distance with unequal coefficients (3x − 4y + 7 and 6x − 8y − 6). Fix: divide the second by 2 first.
Only one line found when the angle condition allows two. Fix: tan θ = ± (m₂ − m₁)/(1 + m₁m₂) gives two slopes.
Final equation left with fractions. Fix: multiply through to the general form with integers.

Test Straight Lines against the clock

Want it against the clock? Take a timed 30-mark chapter test on Straight Lines, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE? Try the JEE-style questions on straight lines and conic sections.

Straight Lines: common questions

What is in CBSE Class 11 Maths Chapter 9, Straight Lines?

The topics are: Slope, Forms of the equation of a line, Distances, Angle between two lines. This page has revision notes for each, and 32 board-style practice questions with step-marked solutions.

How many marks is Straight Lines worth?

It is in the Coordinate Geometry unit, which carries 12 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Straight Lines?

Slope of Ax + By + C = 0 taken as A/B. Fix: rearrange to y = −(A/B)x − C/B. Modulus dropped in the distance formula, giving a negative distance. Fix: |Ax₁ + By₁ + C|/√(A² + B²).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Straight Lines Class 11: notes and important questions

Try it first