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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.
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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.
1. The distance between two points
2. The coordinate of point which cuts a line segment joining by
3. If the ratio of
4. Area of the triangle whose vertices are
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,
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
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
Since the lines are parallel to each other so the angle of inclination is also equal.
Hence,
Taking
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,
Taking
Two lines l1 and l2 are said to be perpendicular if m1m2 = -1
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,
Example 3.4
Find the acute angle between the lines
Solution
Let
Now
Let
Let three points be A(1,4), B(4,6), and C(10,10) lies on the same line.
The slope of the line that passes through the points
The slope of the line that passes through points
So, mAB = mBC =
If the points lie on the same line, then the slope of the line joining any two points is always the same.
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
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
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
Two-point form
Let two points be (x1,y1) and (x2,y2) then
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
Intercept - Form
Let the point on the coordinate axes be (a, 0) and (0, b)
Then the equation
Normal Form
General Equation Of A Line
The general equation goes by
Different forms of
Slope-intercept form
Intercept form
Let P(x1, y1) be a point that does not lie on the line
Then the distance of the point P(x1, y1) from the line is
Distance Between Two Parallel Lines
Suppose the equations of two parallel lines be
Then the distance between two parallel lines is
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:
After finishing the textbook exercises, students can use the following links to check the NCERT exemplar solutions for a better understanding of the concepts.
Students can also check these well-structured, subject-wise solutions.
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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