Straight Lines Class 11th Notes - Free NCERT Class 11 Maths Chapter 10 notes - Download PDF

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Straight Lines Class 11 Notes

Careers360 experts have curated these Straight Lines Class 11 Notes to help students revise quickly and confidently.

Important formulas

1. The distance between two points $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ and $\mathrm{Q}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ is $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

2. The coordinate of the point which cuts a line segment joining by $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ internally in ratio $m:n$ is $\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$

3. If the ratio of $\mathrm{m}: \mathrm{n}=1: 1$ then the coordinate of the point is $\left(\frac{x_2+x_1}{2}, \frac{y_2+y_1}{2}\right)$

4. Area of the triangle whose vertices are $P\left(x_1, y_1\right), Q\left(x_2, y_2\right)$, and $R\left(x_3, y_3\right)$ is given by:
$A=\frac{1}{2}\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|$

Slope of A Line

The steepness of a line is known as the slope of the line.

A line that is nonparallel to the x-axis cuts the x-axis at two angles. The angles are supplementary to each other. A line that makes an angle with the positive direction x-axis and the angle is measured anti-clockwise, then θ is the inclination of the line.

Definition: If a line makes an angle θ with the positive direction x-axis, then tanθ is the slope of the line.

The slope of a line is also denoted by the letter m.

Slope of a line when the coordinates of any two points on the line are given

When two points are given as (x₁, y₁) and (x₂, y₂) the slope is $m=\frac{y_2-y_1}{x_2-x_1}$

Conditions for Parallelism and Perpendicularity of Lines in Terms of Their Slopes

Parallel Lines

Let two lines l₁ and l₂ be parallel to each other. The slopes of lines l₁ and l₂ are m₁ and m_2. The inclinations of the lines l₁ and l₂ are α and β, respectively. The diagram is shown below

Since the lines are parallel to each other so the angle of inclination is also equal.

Hence, $α=β$

Taking $\tan$ both sides

$\tan α = \tan β$ and m₁ = m₂

The slope of line l₁ = The slope of line l₂

If the lines are parallel, then the slopes of the lines are also equal.

Perpendicular Lines

Let two lines l₁ and l₂ be perpendicular to each other. The slopes of lines l₁ and l₂ are m₁ and m₂. The inclinations of the lines l₁ and l₂ are α and β, respectively. The diagram is shown below

So, $β= α+90°$

Taking $\tan$ both sides

$\begin{aligned} & \tan \beta=\tan \left(\alpha+90^{\circ}\right) \\ &⇒ \tan \beta=-\cot \alpha \\ &⇒ \tan \beta=-\frac{1}{\tan \alpha} \\ &⇒ \tan \alpha \cdot \tan \beta=-1 \\ &⇒ m_1 m_2=-1\end{aligned}$

Two lines l₁ and l₂ are said to be perpendicular if m₁m₂ = -1

The Angle Between Two Lines

The inclination of two lines be θ₁ and θ₂ and θ₁, θ₂ ≠ 90°

The slopes of the lines are m₁ = tanθ₁ and m₂ = tanθ₂

Assume that θ is the angle between the lines.
So,
$\begin{aligned} \tan \theta & =\tan \left(\theta_1-\theta_2\right) \\ & =\frac{\tan \theta_1-\tan \theta_2}{1+\tan \theta_1 \tan \theta_2} \\ & =\frac{m_1-m_2}{1+m_1 m_2}\end{aligned}$

Example 3.4
Find the acute angle between the lines $2x-y+3=0$ and $x+y+2=0$
Solution
Let $m_1$ and $m_2$ be the slopes of $2 x-y+3=0$ and $x+y+2=0$
Now $m_1=2, m_2=-1$
Let $\theta$ be the angle between the given lines
$ \tan \theta =\left|\frac{m_1-m_2}{1+m_1 m_2}\right| $
$ \Rightarrow \tan \theta=\left|\frac{2+1}{1+2(-1)}\right|=3 $
$⇒\theta =\tan ^{-1}{(3)}$

Collinearity Of Three Points

Let three points be A(1,4), B(4,6), and C(10,10) lie on the same line.

The slope of the line that passes through the points $A(1,4)$ and $B(4,6)$ is $\frac{6-4}{4-1}=\frac{2}{3}$

The slope of the line that passes through points $B(4,6)$ and $C(10,10)$ is $\frac{10-6}{10-4}=\frac{2}{3}$

So, m_AB = m_BC = $\frac{2}{3}$

If the points lie on the same line, then the slope of the line joining any two points is always the same.

Various Forms of the Equation of a Line

Horizontal and vertical lines

If a line is parallel to the x-axis, and the distance of the line from the x-axis is b units. Then the equation of the line is

Either $y=b$ or $y =-b$.

If a line is parallel to the y-axis, and the distance of the line from the y-axis is a unit. Then the equation of the line is

Either $x=a$ or $x=-a$

Point-Slope Form

Let a fixed point be (x₁,y₁) and an arbitrary point be (x,y).

Let m be the slope of the line.

Then $\frac{y-y_1}{x-x_1}=m$

$y-y_1=m\left(x-x_1\right)$

Two-point form

Let two points be (x₁,y₁) and (x₂,y₂) then

$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Slope-Intercept Form

Let the point on the y-axis be (0,b), and m is the slope of the line

Then the formula is $y=mx+b$

Intercept - Form

Let the points on the coordinate axes be (a, 0) and (0, b)

Then the equation

$\frac{x}{a}+\frac{y}{b}=1$

Normal Form

$x \cos \alpha+y \sin \alpha=p$

General Equation Of A Line

The general equation goes by $Ax+By+C=0$

Different forms of $Ax+By+C=0$

Slope-intercept form

$\begin{aligned} & B y=-A x-C \\ & y=-\frac{A}{B} x-\frac{C}{B} \\ & m=-\frac{A}{B} \text { and } c=-\frac{C}{B}\end{aligned}$

Intercept form

$\begin{aligned} & A x+B y+C=0 \\ &⇒ A x+B y=-C \\ &⇒ \frac{x}{-\frac{C}{A}}+\frac{y}{C}=1 \\ &⇒ \frac{x}{a}+\frac{y}{b}=1 \\ &⇒ a=-\frac{C}{A} \text { and } b=-\frac{C}{B}\end{aligned}$

Distance of a Point from a Line

Let P(x₁, y₁) be a point that does not lie on the line $Ax+By+C=0$.

Then the distance of the point P(x₁, y₁) from the line is

$d=\left|\frac{A x_1+B y_1+C}{\sqrt{A^2+B^2}}\right|$

Distance Between Two Parallel Lines

Suppose the equations of two parallel lines be $Ax+By+C_1 = 0$ and $Ax+By+C_2 = 0$

Then the distance between two parallel lines is $d=\left|\frac{C_1-C_2}{\sqrt{A^2+B^2}}\right|$

Straight Lines: Previous Year Question and Answer

Given below are selected previous year question answers for NCERT Class 11 Maths Chapter 9 Straight Lines, collected from various examinations.

Question 1:
The graph of the equation $x=a(a \neq 0)$ is a_____.

Solution:
Given: $x=a(a \neq 0)$
A diagram that shows the change of one variable to one or more other variables, such as a collection of one or more points, lines, line segments, curves, or regions.
The value of $x$ = The value of $a$ = 1, 2, 3, and so on to infinity.
The value of $y$ = constant (assume $y=4$)
As a result, the graph these points create will be a straight line parallel to the y-axis.
Hence, the correct answer is "a line parallel to the y-axis".

Question 2:
What is the slope of the line parallel to the line passing through the points (5, –1) and (4, –4)?

Solution:
Given: The points are (5, –1) and (4, –4).
$\text{Slope}=\frac{y_2–y_1}{x_2–x_1}$
Substitute the given values in the above formula, and we get,
$\frac{–4–(–1)}{4–5}=\frac{–3}{–1}$
So, the slope of the given line is 3. The slope of the line parallel to this line is also the same.
Hence, the correct answer is $3$.

Question 3:
At what point does the line $2x–3y = 6$ cut the Y-axis?

Solution:
Given:
The equation is $2x–3y = 6$
The Y-axis can be expressed as $x = 0$.
So, both lines should be satisfied at the intersection.
Substitute $x=0$ in the equation, $2x–3y = 6$, we get,
$(2×0)–3y = 6$
⇒ $y = –2$
The point is (0, –2), which satisfies both lines.
Hence, the correct answer is (0, –2).

Importance of NCERT Class 11 Maths Chapter 9 Notes

NCERT Class 11 Maths Chapter 9 Notes play a vital role in helping students grasp the core concepts of the chapter easily and effectively, so that they can remember these concepts for a long time. Some important points of these notes are:

• Effective Revision: These notes provide a detailed overview of all the important theorems and formulas, so that students can revise the chapter quickly and effectively.
• Clear Concepts: With these well-prepared notes, students can understand the basic concepts effectively. Also, these notes will help the students remember the key concepts by breaking down complex topics into simpler and easier-to-understand points.
• Time Saving: Students can look to save time by going through these notes instead of reading the whole lengthy chapter.
• Exam Ready Preparation: These notes also highlight the relevant content for various exams, so that students can get the last-minute guidance for exams.

NCERT Class 11 Maths Notes – Chapter-Wise Links

For students' preparation, Careers360 has gathered all Class 11 Maths NCERT Notes here for quick and convenient access.

Subject-Wise NCERT Exemplar Solutions

After finishing the textbook exercises, students can use the following links to check the NCERT exemplar solutions for a better understanding of the concepts.

• NCERT Exemplar Class 11 Solutions
• NCERT Exemplar Class 11 Maths
• NCERT Exemplar Class 11 Physics
• NCERT Exemplar Class 11 Chemistry
• NCERT Exemplar Class 11 Biology

Subject-Wise NCERT Solutions

Students can also check these well-structured, subject-wise solutions for class 11.

• NCERT Solutions for Class 11 Mathematics
• NCERT Solutions for Class 11 Chemistry
• NCERT Solutions for Class 11 Physics
• NCERT Solutions for Class 11 Biology

NCERT Books and NCERT Syllabus

Before the start of a new academic year, students should refer to the latest syllabus to determine the chapters they’ll be studying. Below are the updated syllabus links, along with some recommended reference books.

• NCERT Books Class 11 Maths
• NCERT Syllabus Class 11 Maths
• NCERT Books Class 11
• NCERT Syllabus Class 11