Straight Lines Class 11
Coordinate geometry refers to the field of mathematics that deals with a geometrical study using algebra. A French mathematician Rene Descartes realised that representation of a curve or a straight line is possible using an algebraic equation. In this field of mathematics, an ordered pair of numbers (real numbers) represents a point in a plane and is known as coordinates of that point. Also, a curve or a straight line is illustrated using an algebraic equation having real coefficients.
Notably, this is one of the most important chapters since numerous other functions and calculations also depend on it.
What is Meant by a Straight Line?
A straight line is a line traced by a point that is travelling in a fixed direction having zero curvature. In simpler words, straight lines can also be said as the shortest paths between two points.
Straight Line: Slope
Inclination
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When a straight line L makes an angle with +ve x-axis which is evaluated in anticlockwise direction till the portion of the line L above the x-axis is known as angle of inclination or just inclination of the line. Usually, it is denoted by theta (θ). According to the figure shown above, it is evident that theta is higher than equal to 0 degrees and less than 180 degrees.
Among Vital Pointers to Note,
The angle of inclination of any line which is parallel to the x-axis or x-axis itself is 0 degrees.
The angle of inclination of any line which is parallel to the x-axis or y-axis itself is 90 degrees.
Vertical, Horizontal and Oblique Lines
A line that is parallel to the x-axis or x-axis itself is termed a straight horizontal line.
A line that is parallel to the y-axis or y-axis itself is termed as a vertical straight line.
A line that is neither vertical nor horizontal is termed an oblique line.
Slope or Gradient of a Straight Line
If theta (not equal to 90 degrees) is the angle of inclination of all straight lines, then their slope or gradient is tan θ. It is represented by “m”.
Therefore, it can be said that when θ (not equal to 90 degrees) is an angle of inclination, m is equal to tan θ.
Observations
As tan θ is not determined when θ is 90 degrees, a vertical line’s slope is also not determined.
Y-axis slope is not determined.
The angle of inclination of all parallel lines to the x-axis is 0 degrees, so the slope is tan 0 degrees which is equal to 0. Hence, every horizontal line’s slope is 0.
X-axis slope is 0.
How to Find a Straight Line’s Slope Which Joins Two Points?
The slope in a straight line (which is non-vertical) moving through two points can be determined by the following expression.
Consider P(x1, y1) and Q(x2, y2) are two points online l, which is non-vertical. As the line is non-vertical, then x2 is not equal to x1.
Let the angle of inclination of line l be θ, and it can be obtuse or acute. Both cases are required to be considered. From Q and P, perpendiculars QN and PM on x-axis and PL perpendicular to NQ are drawn, which is shown the following figures:
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Case 1. When θ is acute:
PL= MN= ON – OM = x2 – x1 and LQ= NQ – NL = NQ – MP = y2 – y1
So, angle QPL = θ
Therefore, tan θ = LQ/PL = y2 –y1 /x2 –x1
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Case 2. When θ is obtuse:
LP= NM = OM-ON = x1 – x2 and LQ= NQ – NL = NQ – MP = y2 – y1
So, angle QPL = π – θ
Therefore, tan (π – θ) = LQ/LP = y2 – y1 /x1 – x2
-tan θ = y2 – y1/x1 – x2
tan θ = y2 – y1 / x2 – x1
So, for both the cases, slope m is same.
The above concise information related to straight lines will be able to clear your concept about the slope of a straight line. As this chapter comprises various line equations and types of straight lines, you must refer to our online study materials and download the Vedantu app for online classes.
Properties of a Straight Lines
Below are the properties of a straight line:
Since a straight line has infinite length, you can never calculate the distance between the two endpoints of that line.
Every straight line is a unidimensional figure that has no thickness and can extend infinitely in both directions.
Straight lines do not have an area or volume, only an infinite length.
An infinite number of straight lines can pass through one point but there will only be one unique line that passes through two points.
Different Types of Straight Lines
Straight lines are classified into different types based on their alignment, which refers to the angle they create with either of the axes on a graph. Let’s learn more about these types of straight lines.
Horizontal Straight Line
Lines that are perpendicular to the y-axis but parallel to the x-axis in a graph are known as horizontal straight lines. These lines form a 90-degree angle with the y-axis and a 0-degree angle with the x-axis.
Vertical Straight Line
As opposed to horizontal lines, vertical straight lines are perpendicular to the x-axis but parallel to the y-axis. Similarly, they form a 90-degree angle with the x-axis and a 0-degree angle with the y-axis.
Oblique Straight Line
Also known as slanted lines, Oblique lines are the ones that are neither horizontal nor vertical in alignment. These lines can form any angle, except 0, 90, 180, 270, and 360 degrees, with both axes.
Important points about Straight Lines
When you start solving questions based on Straight Lines – Meaning, Observation and Cases, you have to keep the following points in mind:
A straight line will never pass through three non-collinear points.
If two lines, say A and B, coincide, they follow the relation: A = c × B, where c is a real number.
The formula to calculate the acute angle, say θ, between two straight lines having slope M1 and M2, where M1 < M2, is:
Tan θ = (M2-M1)/(1 + M2 × M1)
How To Learn the Concept of Straight Lines – Meaning, Observation and Cases
Learning the concept of Straight Lines will be easier than you think. It might feel a little tricker at first but once you start solving questions using the equations, you will get a hold of it. Firstly, you must have all the study materials, including revision notes, NCERT solutions, reference books, etc., to start study. These study resources make the learning process easy and smooth. After that, you can use the following tips to understand this crucial concept:
Before you explore the practical part of Straight Lines, you should go through the definition of straight lines and all their types to get an idea of what the concept is about actually.
Use your textbook to read explanations of topics covered under Straight Lines. In your textbook, you will find detailed explanations along with pictorial representations that will give you a better idea of the concept.
Once you have read the theoretical part, you should start with the equation of straight lines and understand how to use this equation in different types of questions.
Before solving any questions, go through the examples based on Straight Lines to understand the right sequence of steps to solve a problem.
Once you are done with Straight Lines, go through your revision notes to quickly revise what you have learned.
To explore more questions, you can use reference books, sample papers, and previous year questions papers.
You can also use sample papers and previous year question papers to get an idea of the difficulty level of the questions, that can be asked in your exam.
FAQs on Straight Lines
1. What is the Slope of a Line Moving through Points (i) (3, -2) and (-1, 4) (ii) (3, -2) and (7, -2)?
(i) By substituting the values in the equation, the slope of line moving through (3, -2) and (-1, 4) is
= (4 -(- 2)) / (-1 – 3)
= 6 / -4
= -3 / 2
(ii) Now for the points (3,-2) and (7, -2) the slope is:
= (-2 -(-2)) / (7-3)
= 0 / 4
= 0
2. What is the Value of k if Three Straight Lines 2x + y – 3= 0, 5x + ky -3 =0 and 3x – y – 2=0 are Concurrent?
Since the three lines are said to be concurrent, the intersecting point of two lines falls on the third line.
Let the given equations be marked as:
2x + y -3 = 0.........(1)
5x + ky – 3 = 0 ……(2)
3x – y – 2 = 0 ……..(3)
Solving equations 1 and 3 by cross multiplication procedure:
x/-2-3 = y/ -9+4 = 1/-2-3 or x=1 and y=1
Hence, the intersecting point of the two straight lines is (1,1). Now, by putting the values in equation 2:
5 . 1 + k . 1 – 3 = 0 or k = -2
The value of k is -2
3. What will be the Straight-line Equation of Three Points (5,1), (1,-1) and (11,4)? Also, Prove that these Points on the Straight Line are Collinear.
Let the mentioned points be P(5,1), Q(1, -1) and R(11,4). So, the line equation is:
y-1 = (-1 -1/1-5) (x-5)
y-1 = (-2/-4) (x-5)
y-1 =1 /2 (x-5)
2(y-1) = (x-5)
2y -2 = x-5
x- 2y – 3 =0
Now it is seen that R(11,4) satisfies x – 2y – 3 = 0. Therefore it can be said that all the points lies on the same line and they are collinear.
4. What will be the Fourth Vertex if Three Back to Back Vertices of a Parallelogram are (-2,-1), (1,0) and (4,3)?
Consider the vertices to be A(-2,-1), B(1,0), C(4,3) and D(x,y). Then the mid-point of AC will be (-2+4/2, -1+3/2) which is (1,1).
Moreover, the mid-point of BD will be ((1+x)/2, (0+y)/2).
As the parallelogram diagonals bisect each other, the midpoints are the same.
(1+x)/2 =1 and (0+y)/2 =1
Which means x = 1 and y = 2
Therefore the fourth vertex is (1,2).
5. Which platform is the best to learn the concept of Straight Lines?
When it comes to learning, Vedantu is the best platform for you. If you want to learn the concept of Straight Lines, you can just visit Vedantu.com or download our mobile application from the app store or the play store. To start learning on Vedantu, you don’t have to pay for anything as our content is available to download for absolutely free. Apart from the Straight Lines, you can learn other important topics of Maths as well. So, whether you want to learn trigonometry or integrals, Vedantu is the best place for you.
