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

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

Edited By Komal Miglani | Updated on Apr 07, 2025 09:15 PM IST

If you are told to go from one end of a cricket ground to the other end, what would you do? You would generally take the shortest path, right? Well, have you noticed that the shortest path between two points is always a straight line? Straight lines are one of the most fundamental parts of geometry, which play a significant role in navigation, architecture, road construction, etc. From NCERT Class 11 Maths, the chapter Straight Lines contains the concepts of Slope of a line, Angle between two lines, Different forms of the equation of a line, Distance of a point from a line, etc. Understanding these concepts will help the students grasp more advanced trigonometry topics easily and will also enhance their problem-solving ability in real-world applications.

This Story also Contains
  1. NCERT Class 11 Maths Chapter 9 Notes
  2. Importance of NCERT Class 11 Maths Chapter 9 Notes
  3. NCERT Class 11 Notes Chapter Wise
  4. Subject-Wise NCERT Solutions
  5. NCERT Books and Syllabus

This article on NCERT notes Class 11 Maths Chapter 9 Straight Lines offers well-structured NCERT notes to help the students grasp the concepts of Straight Lines easily. Students who want to revise the key topics of Straight Lines quickly will find this article very useful. It will also boost the exam preparation of the students by many folds. These notes of NCERT Class 11 Maths Chapter 9 Straight Lines are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 11 maths and NCERT solutions for other subjects and classes can be downloaded from the NCERT Solutions.

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NCERT Class 11 Maths Chapter 9 Notes

Important formulas:

1. The distance between two points P(x1,y1) and Q(x2,y2) is d=(x2x1)2+(y2y1)2
2. The coordinate of point which cuts a line segment joining by P(x1,y1) and Q(x2,y2) internally in ratio m:n is (mx2+nx1m+n,my2+ny1m+n)
3. If the ratio of m:n=1:1 then the coordinate of the point is (x2+x12,y2+y12)
4. Area of the triangle whose vertices are P(x1,y1),Q(x2,y2), and R(x3,y3) is given by:
A=12|x1(y2y3)+x2(y3y1)+x3(y1y2)|

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 to two angles. The angles are supplementary of 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 slope of the line.

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

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Slope of a line when 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=y2y1x2x1

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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 m2. The inclinations of the lines l1 and l2 are α and β, respectively. The diagram is shown below

1647348048971

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

Hence, α=β

Taking tan both sides

tanα=tanβ 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

1647412033627

So, β=α+90°

Taking tan both sides

tanβ=tan(α+90)tanβ=cotαtanβ=1tanαtanαtanβ=1m1m2=1

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

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Assume that θ is the angle between the lines.
So,
tanθ=tan(θ1θ2)=tanθ1tanθ21+tanθ1tanθ2=m1m21+m1m2

Example 3.4
Find the acute angle between the lines 2xy+3=0 and x+y+2=0
Solution
Let m1 and m2 be the slopes of 2xy+3=0 and x+y+2=0
Now m1=2,m2=1
Let θ be the angle between the given lines
tanθ=|m1m21+m1m2|=|2+11+2(1)|=3θ=tan1(3)

Collinearity Of Three Points

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

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The slope of the line that passes through the points A(1,4) and B(4,6) is 6441=23

The slope of the line that passes through points B(4,6) and C(10,10) is 106104=23

So, mAB = mBC = 23

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

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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 yy1xx1=m

yy1=m(xx1)

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Two-point form

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

yy1=y2y1x2x1(xx1)

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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

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Intercept - Form

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

Then the equation

xa+yb=1

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Normal Form

xcosα+ysinα=p

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General Equation Of A Line

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

Different forms of Ax+By+C=0

Slope-intercept form

By=AxCy=ABxCBm=AB and c=CB

Intercept form

Ax+By+C=0Ax+By=CxCA+yC=1xa+yb=1a=CA and b=CB

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=|Ax1+By1+CA2+B2|

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Distance Between Two Parallel Lines

Suppose the equations of two parallel lines be Ax+By+C1=0 and Ax+By+C2=0

Then the distance between two parallel lines is d=|C1C2A2+B2|

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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 contents for various exams, so that students can get the last-minute minute very useful guidance for exams.
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NCERT Class 11 Notes Chapter Wise

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.

Subject-Wise NCERT Solutions

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

NCERT Books and Syllabus

Students should always analyze the latest CBSE syllabus before making a study routine. The following links will help them check the syllabus. Also, here is access to more reference books.

Happy learning !!!

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

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0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

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zero\;

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K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

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be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

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Weight fraction of solute

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Fraction of solute present in water

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Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

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half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

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more than 9

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