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Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce PDF Download

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Introduction

A line is considered as a geometrical shape with no breadth. It extends in both directions with no endpoints. It is a set of points and only has length. Lines can be parallel, perpendicular, intersecting or concurrent. A line in a coordinate plane forms two angles with the x-axis, which are supplementary.  The angle (say) θ made by the line l with the positive direction of the x-axis and measured anti-clockwise is called the inclination of the line. Thus 0° ≤ θ ≤ 180°.

Slope of a Line

  • The basics of straight lines start with a slope. A slope is an inclined position. It forms a certain angle with the base. 
  • How can we find a slope of a line? In the coordinate geometry, if any line l makes an angle θ with the positive direction of x-axis, it is called the Inclination of the line.
  • The angle is measured in an anti-clockwise way. A slope of a line is the tangent of the inclination, θ i.e., tan θ. It is denoted as m.
    Thus, the slope of a line:
    m = tan θ, θ ≠ 90°.

Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce

Slope of a Line When Coordinates of Any Points on the Lines Are Given

  • Imagine a line l with a slope θ. Two points A (x1, y1) and B (x2, y2) lies on it. The angle of inclination can be acute or obtuse.

(i) Case 1: θ is acute

Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce


Here, ∠CAB = θ. The slope of the line, m = tan θ.
In ΔCAB, tan θ = CB/CA = (y2 − y1)/(x2 − x1).
Thus, m = tan θ = (y2 − y1)/(x2 − x1).


(ii) Case 2: θ is obtuse

Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce


Here, ∠CAB =180° − θ.Slope of the line, m = tan θ = tan (180°− ∠CAB) = − tan ∠CAB
m = − CB/CA = − (y2 − y1)/(x1 − x2).
Thus, m = tan θ = (y2 − y1)/(x2 − x1).

Conditions for Parallelism of Lines in Terms of Slopes

Two lines are parallel if the distance between them at any point remains the same. It can also be inferred that the slopes of the two lines must be the same. Let two lines land l2 have respective slopes m1 and m2 and angle of inclinations α and β. The lines will be parallel if α = β i.e., m1 = m2 and tanα = tanβ

Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce


Conditions for Perpendicularity of Lines in Terms of Slopes

Two lines are perpendicular if they intersect each other at an angle of 90°. Let two lines l1 and l2 have respective slopes m1 and m2 and angle of inclinations α and β. Here, α = β + 90°.

Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce

tan α= tan (β + 90°) = − cot β = − 1/tan β

or,  m2 = −1 /m1 or m1m2 = −1

The lines will be perpendicular if and only if m2 = −1/m1 or m1m2 = −1.

 Angle Between Two Lines
Above we get to know about parallel and perpendicular lines. How can we find out the angle between two intersecting lines (other than 90°)? Let two lines l1 and l2 have respective slopes m1 and m2 and angle of inclinations α1 & α2. Or, m1= tan α1 & m2 = tan α2

Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce

From the property of angle:

θ =  α1 − α2

⇒ tan θ = tan (α1 − α2) = (tanα1 − tanα2)/ (1+ tanα1 tanα2) = (m1−m2)/ (1+m1m2)

and Φ = 180° − θ,

⇒ tan Φ = tan (180° − θ) = − tan θ = − (m1−m2) / (1+m1m2)

1 + m1m2 ≠ 0

Collinearity of Three Points

Three points are collinear if they all lie on the same line. Three points A(x1,y1), B(x2,y2) and C(x3,y3) are collinear iff slope of AB = slope of BC i.e., (y2 − y1)/(x2 − x1) = (y3 − y2)/(x3 − x2).

Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce

Solved Examples

Problem 1. What is the slope of the horizontal line and vertical line
Solution: The slope of horizontal line is zero (m = 0, θ = 0°). The slope of a vertical line is undefined (θ= 90°).

Problem 2. Find the slope of the line passing through the points (4, 3) and (1, 5).

Solution: Slope of the line, m = (y2 − y1)/(x2 − x1) = (5 − 3) / (1 − 4) = −2/3. 

Problem 3. Find the value of x, if the points (1, −1), (x, 1) and (6, 7) are collinear.

Solution: Three points are collinear if (y2 − y1)/(x2 − x1) = (y3 − y2)/(x3 − x2) . Putting the values, we have, (1 − (−1))/(x − 1) = (7 − 1)/(6 − x) or, 2(6 − x)  = 6(x − 1). Or, x = 9/4.

The document Basics of Straight Lines | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Basics of Straight Lines - Mathematics (Maths) Class 11 - Commerce

1. What are the basics of straight lines in Airforce X Y/Indian Navy SSR?
Ans. The basics of straight lines in Airforce X Y/Indian Navy SSR refer to the fundamental concepts and properties related to straight lines. This includes understanding the slope, intercepts, and equations of straight lines, as well as the various forms of equations such as point-slope form, slope-intercept form, and general form.
2. How are straight lines related to the Airforce X Y/Indian Navy SSR exam?
Ans. Straight lines are an important topic in the Airforce X Y/Indian Navy SSR exam as they form the foundation for many mathematical concepts and calculations. Understanding the basics of straight lines is crucial for solving problems in geometry, trigonometry, and other related topics that are part of the syllabus for the exam.
3. What are some examples of questions related to straight lines in the Airforce X Y/Indian Navy SSR exam?
Ans. Some examples of questions related to straight lines in the Airforce X Y/Indian Navy SSR exam could be: 1. Find the equation of a line passing through two given points. 2. Determine the slope of a line parallel to a given line. 3. Calculate the distance between a point and a line. 4. Find the x-intercept and y-intercept of a given line. 5. Solve a system of linear equations involving two lines.
4. How can I prepare effectively for the straight lines topic in the Airforce X Y/Indian Navy SSR exam?
Ans. To prepare effectively for the straight lines topic in the Airforce X Y/Indian Navy SSR exam, you can follow these steps: 1. Understand the basic concepts and properties of straight lines. 2. Practice solving a variety of problems related to straight lines. 3. Familiarize yourself with the different forms of equations for straight lines. 4. Review and revise any formulas or theorems related to straight lines. 5. Solve previous year question papers and mock tests to assess your preparation level.
5. Are there any specific tips or tricks for solving straight line problems in the Airforce X Y/Indian Navy SSR exam?
Ans. Yes, here are some tips and tricks for solving straight line problems in the Airforce X Y/Indian Navy SSR exam: 1. Draw accurate diagrams or graphs whenever possible to visualize the problem. 2. Use the appropriate form of equation based on the given information. 3. Remember the special cases, such as vertical lines with undefined slope or horizontal lines with zero slope. 4. Use the concept of symmetry to simplify calculations in certain cases. 5. Practice mental math techniques to quickly calculate slopes, intercepts, and distances.
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