Cbse Class 11 Maths Notes Chapter 9 Straight Lines

Section–A (1 Mark Questions)

1. A line passing through the point (1,2) and having a slope of 4 intersects the y-axis at(0,b). Find the value of b .

Ans. $$\begin{aligned}& \text { } \because m=4 \\& \Rightarrow \frac{b-2}{0-1}=4 \\& \Rightarrow b-2=-4 \\& \therefore b=-2 .\end{aligned}$$

2. Find the equation of the line which cuts off an intercept 4 on the positive direction of x-axis and an intercept 3 on the negative direction of y-axis.

Ans. Given, $a=4$ and $b=-3$

Equation of line is given by

$$\begin{aligned}& \frac{x}{a}+\frac{y}{b}=1 \\& \Rightarrow \frac{x}{4}+\frac{y}{(-3)}=1 \\& \Rightarrow-3 x+4 y=-12 .\end{aligned}$$

3. Find the equation of the line which is at a distance 3 from the origin and the perpendicular from the origin to the line makes on angle of $30^{\circ}$ with positive direction of x-axis

Ans. Given, $p=3$ and $\omega=30^{\circ}$

Equation of line is given by

$$\begin{aligned}& x \cos \omega+y \sin \omega=p \\& \Rightarrow x \cos 30^{\circ}+y \sin 30^{\circ}=3 \\& \Rightarrow x\left(\frac{\sqrt{3}}{2}\right)+y\left(\frac{1}{2}\right)=3 \\& \therefore \sqrt{3} x+y=6 .\end{aligned}$$

4. Find the value of a for which the line ax+2y+3=0 has a slope of 5.

Ans. Given line: $a x+2 y+3=0$

$$\begin{aligned}& \Rightarrow 2 y=-a x-3 \\& \Rightarrow y=\left(-\frac{a}{2}\right) x+\left(-\frac{3}{2}\right)\end{aligned}$$

Thus, $m=-\frac{a}{2}=5$

$$\therefore a=-10 \text {. }$$

5. Find the mid-point of the line segment joining the points (-3,2) and (7,4).  

Ans. Mid-point of the line segment joining the points $(-3,2)$ and $(7,4)$ is

$$\left(\frac{-3+7}{2}, \frac{2+4}{2}\right)=(2,3) \text {. }$$

Section–B (2 Marks Questions)

6. Find the angle between x-y=2 and x-3y=6.

Ans. Given lines: $x-y=2$ and $x-3 y=6$

Slopes are 1 and $\frac{1}{3}$.

Angle between the lines:

$$\begin{aligned}& \tan \theta=\left|\frac{\frac{1}{3}-1}{1+\left(\frac{1}{3} \times 1\right)}\right|=\left|\frac{\frac{-2}{3}}{\frac{4}{3}}\right| \\& \therefore \theta=\tan ^{-1}\left(\frac{1}{2}\right) .\end{aligned}$$

7. Find the slope of the line passing through the point (-3,6) and the middle point of the line joining the points (4,-5) and (-2,9). 

Ans. Mid-point of the line segment joining the points $(4,-5)$ and $(-2,9)$ is

$$\left(\frac{4-2}{2}, \frac{-5+9}{2}\right)=(1,2)$$

Slope of the line passing through the points $(1,2)$ and $(-3,6)$ is

$$m=\frac{6-2}{-3-1}=\frac{4}{-4}=-1 \text {. }$$

8. Equation of the line passing through (-3,-2) and having y-intercept of 2 units is ………

Ans. Given: $y$-intercept $=2$

i.e., line passes through the point $(0,2)$.

Equation of line passing through $(-3,-2)$ and $(0,2)$ is:

$y-(-2)=\frac{2-(-2)}{0-(-3)}(x-(-3))$

$\Rightarrow y+2=\frac{2+2}{3}(x+3)$

$\Rightarrow 3 y+6=4 x+12$

$\Rightarrow-4 x+3y=6$

9. Two lines 2x+3y+4=0 and 3x-2y+1=0 are ………. to each other.  (Parallel/Perpendicular).  

Ans. Given,

$$\begin{aligned}& 2 x+3 y+4=0 \\& 3 x-2 y+1=0\end{aligned}$$

Slope of line (i) is $-2 / 3$

Slope of line (ii) is $3 / 2$

$\because$ Slope of line (i) $\neq$ Slope of line (ii)

$\therefore$ lines are not parallel.

Now, product of slopes $=\left(-\frac{2}{3}\right) \times\left(\frac{3}{2}\right)=-1$ i.e., lines are perpendicular.

10. Find the angle between the lines having inclination of $30^{\circ}$ and 45^{\circ} with positive direction of x-axis. 

Ans. 

$$ \text { } \begin{aligned} & m_1=\tan 30^{\circ}=\frac{1}{\sqrt{3}} \\ & m_2=\tan 45^{\circ}=1 \\ & \tan \theta=\left|\frac{m_2-m_1}{1+m_1 m_2}\right| \\ &=\left|\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}\right|=\frac{\sqrt{3}-1}{\sqrt{3}+1} \\ &=\left|\frac{(\sqrt{3}-1)^2}{3-1}\right|=\left|\frac{3+1-2 \sqrt{3}}{2}\right| \\ &=\left|\frac{4-2 \sqrt{3}}{2}\right|=|2-\sqrt{3}| \\ & \therefore \theta=\tan ^{-1}|2-\sqrt{3}| . \end{aligned} $$

11. Find the coordinates of the point of intersection of the lines 2x-y+3=0 and x+2y-4=0. 

Ans. 2x-y+3=0…. (i)

 x+2y-4=0…. (ii)

Solving equation (i) and (ii), we get, x=-2/5 and y=11/5

Hence point of intersection of given lines is $\left ( \frac{-2}{5},\frac{11}{5} \right )$ .

12. Reduce the equation $\sqrt{3}+y+2=0$ to the normal form and find p and $\omega$  .

Ans. $\sqrt{3} x+y+2=0$

13. Put the equation $\frac{x}{a}+\frac{y}{b}=1$ to the slope intercept form and find its slope and y-intercept.

Ans. Given, $\frac{x}{a}+\frac{y}{b}=1$

$$\begin{aligned}& \Rightarrow \frac{y}{b}=-\frac{x}{a}+1 \\& \Rightarrow y=\left(-\frac{b}{a}\right) x+b \\& \therefore m=-\frac{b}{a} \text { and } c=b\end{aligned}$$

PDF Summary - Class 11 Maths Straight Lines (Chapter 10)

Class 11 Maths Notes of Straight Lines helps you to understand all the important concepts related to the straight line. These revision notes are prepared by our subject experts as per the latest CBSE syllabus and any changes that have been made in the syllabus are taken into consideration. Maths Class 11 Straight Lines Notes are available in free pdf format that you can easily download and access it anywhere and anytime.

These Straight Lines Class 11 notes are prepared by subject experts at Vedantu with close to 20 years of experience in teaching Mathematics after reviewing the last 10-year question papers. This helps us to offer short, precise, and productive Class 11 Maths Notes of Straight Lines for the Class 11 students. Download free Maths Class 11Straight Lines Notes pdf with just a single click on the pdf link given below.

A Quick Glimpses of  Class 11 Maths Chapter 10 Straight Lines

In Class 11 Maths Chapter 10 Straight Lines, the basic concepts of lines such as slope, the angle between two lines, different forms of lines, and the distance between lines are described in detail. Class 11 Revision Notes Straight Lines include important topics along with the formula for the students so that they can learn and prepare for the exams accordingly. 

These revision notes will surely help students to score well in exams. Read the article below to get further information about Class 11 Chapter 10.

What is a Straight Line?

A straight line is defined as a line drawn up by the points traveling in a constant direction with zero curvature. In other words, we can say that the straight line is the shortest distance between two points.

General Form of a Line

The relation between variables such as x and y agrees with all points on the curve.

The general form of the equation of a straight line is given as:

Ax + By + C = 0

Where, A, B, and C are constants and x, y are variables.

Slope of a Line

Tan θ is known as the slope or gradient of a line L if θ is the gradient of point L. The slope of the line is the line whose inclination is not equal to 90 degrees.

Hence, M = Tan θ, and θ is not equal to 90°.

It is seen that the slope of the x- axis is 0, if the slope of the y-axis is not defined.

Slope Intercept Form

The straight - line equation  in slope-intercept form is given as:

Y = mx + C

Where m represents the slope of the line and C is the y-intercept.

Shortest Straight Line Distance

The shortest straight line distance between two points say P and Q having coordinates (P₁, Q₁) and (P₂, Q₂) is expressed as:

\[PQ = \sqrt{(P_{1} - Q_{1})^{2} + (P_{2} - Q_{2})^{2}}\]

Important Questions from Straight Lines (Short, Long, and Practice Questions)

Short Answer Type Questions

1. Equation of the line passing through (0, 0) and slope m is ____.

2. The angle between the lines x – 2y = y and y – 2x = 5 is ____.

3. Find the measure of the angle between the lines x+y+7=0 and x-y+1=0.

Long Answer Type Questions

1. Find the equation of the locus of a point equidistant from the point A(1, 3) to B(-2, 1).

2. Find y-intercept of the line 4x – 3y + 15 = 0.

3. If two vertices of a triangle are (3, -2) and (-2, 3) and its orthocenter is (-6, 1), then find its third vertex.

Practice Questions

1. Find the measure of the angle between the lines x+y+7=0 and x−y+1=0.

2. Find the equation of the line that has y−intercept 4 and is perpendicular to the line y=3x−2.

3.  Equation of a line is 3x−4y+10=0 find its slope.

4. Find the slope of the line, which makes an angle of 30o with the positive direction of y−axis measured anticlockwise.

Why Choose Vedantu’s Class 11 Maths Notes of Chapter 10 Straight Lines?

Class 11 Maths Notes of Chapter 10 Straight Lines offered by Vedantu provide many benefits to the students who are preparing for the Chapter 10 Straight line. With the help of Straight Lines Class 11 notes, students will be able to revise all the important topics discussed in the chapter quickly without wasting much time.

They will be able to learn all the important concepts and equations and can also remember what they have studied in this chapter as these revision notes will provide brief information on the topics discussed in the chapter. Students can easily access these revision notes anytime and anywhere and can have a strong command of all the topics discussed in the chapter before the exam.

Key Features of Revision Notes for Class 11 Maths Chapter 10 - Straight Lines

• All the points are curated as per the examination point of view to help students score better.

• Concepts are explained in a step-by-step manner.

• These Revision Notes are easy to understand and learn as they are clearly written by subject experts in easy-to-understand language. 

• Explained all concepts that are mentioned in the curriculum.

• These Revision Notes for Class 11 Maths Chapter 10 - Straight Lines help in developing a good conceptual foundation for students, which is important in the final stages of preparation for board and competitive exams.

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Conclusion

The Class 11 CBSE Maths Chapter 10 on "Straight Lines" is a fundamental and important topic in the study of geometry and algebra. The chapter introduces students to the concept of lines, their equations, and various properties associated with them. Students learn how to find the slope, intercepts, and angles between lines. Additionally, they explore the parallel and perpendicular lines and their equations. The chapter also covers the distance formula and the area of a triangle formed by three points. Understanding straight lines is crucial for higher-level mathematical concepts and real-world applications. By mastering this chapter, students build a solid foundation for advanced mathematics.

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