Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS
Introduction
The point-slope form of a straight line equation is useful when we know the slope of the line and a point through which it passes. This form is widely used in coordinate geometry to determine the equation of a line quickly.
Graphical Representation
The point-slope form helps in quickly sketching a straight line when given a point and its slope. Slope Definition
The slope defines the steepness of the line. The word steepness means how much the line is slanted. In other words, the slope shows the direction of a line on the coordinate plane. It is also known as the gradient of a line.
The real-life example of the slope is a ramp, stairs, and sliding swing, etc.
Table of Contents
Point-Slope form of Straight Line Equation
Let any straight line on the two-dimensional coordinate plane and having its slope is m and (x1, y1) be a fixed point lying on the line. Let (x, y) be any arbitrary point on the line.
Since the slope of the line is m, i.e. slope between any two points lying on the line will always be m.
The Slope of Straight Line be,
m = (change in y Co-ordinate) / (change in x Co-ordinate)
So derive the equation of any straight line with a given slope and passing through a fixed point (x1, y1).
Since (x, y) and (x1, y1) are two points on the line and m is the slope of the straight line, slope between (x, y) and (x1, y1) will be equal to m.
Now, slope between (x, y) and (x1, y1) = m
m = (y – y1) / (x – x1)
∴ (y − y1) / (x − x1) = m …(1)
Multiplying both sides of equation (1) by (x − x1) we get,
⟹ (y − y1) = m (x − x1) …(2)
which is the equation of straight line or the point-slope form.
The equation (2) only contains the slope (m) and the fixed point (x1, y1) on the straight line as constants, so it is called point-slope form.
The Point Slope Form Formula is,
| y – y1 = m (x – x1) |
Where,
- m is the slope of the line
- x1 is the coordinates of the x-axis
- y1 is the co-ordinates of the y-axis
Sample Problems on Point-Slope Equation
Problem 1: Find the equation of the straight line with slope 3 and passing through point (-1, 5).
Solution:
Given slope of line is 3 or m = 3 and the line passes through fixed point (-1, 5), or (x1, y1) = (-1, 5). Let (x, y) be any point on the line
The point-slope form of straight line is ,
(y − y1) = m (x − x1)
Putting m = 3 and (x1, y1) = (-1, 5) we get,
⇒ y – 5 = 3(x – (-1))
⇒ y – 5 = 3(x + 1)
⇒ y – 5 = 3x + 3
⇒ y – 3x – 5 – 3 = 0
⇒ y – 3x – 8 = 0,
which is the required equation
Problem 2: Find the equation of the straight line with slope -2 and passing through (7, -4).
Solution:
Given slope of the line is -2 or m = -2 and the line passes through fixed point (7, -4), or (x1, y1) = (7, -4). Let (x, y) be any point on the line.
The point-slope form of straight line is ,
(y − y1) = m (x − x1)
⇒ y + 4 = (-2)(x – 7)
⇒ y + 4 = -2x + 14
⇒ y + 2x + 4 – 14 = 0
⇒ y + 2x – 10 = 0,
which is the required equation
Problem 3: Find the equation of the straight line with slope 1/4 and passing through (2, 3).
Solution:
Given slope of the line is 1/4 or m = 1/4 and the line passes through fixed point (2, 3) or (x1, y1) = (2, 3). Let (x, y) be any point on the line.
The point-slope form of straight line is ,
(y − y1) = m (x − x1)
⇒ (y – 3) = (1/4)(x – 2)
Multiplying both sides by 4 we get,
⇒ 4(y – 3) = 1(x – 2)
⇒ 4y – 12 = x – 2
⇒ 4y – x – 12 + 2 = 0
⇒ 4y – x – 10 = 0,
which is the required equation
Problem 4: Find the equation of a line which passes through the point (2, 6) and has a slope of 7.
Solution:
Given, m = 7
(x1, y1) = (2, 6)
The point slope form formula is,
y – y1 = m(x – x1)
y – 6 = 7(x – 2)
y – 6 = 7x – 14
7x – y – 8 = 0
The equation of the line is: 7x – y – 8 = 0
Problem 5: Write the point slope equation of a line with slope 5 that passes through the point (3, -2).
Solution:
Given point is (x1, y1) = (3, -2)
Slope = m = 5
Point slope equation is:
y – y1 = m(x – x1)
y – (-2) = 5(x – 3)
This can be further simplified as:
y + 2 = 5x – 15
5x – y – 15 – 2 = 0
5x – y – 17 = 0
Problem 6: Find the slope and the point which the equation y – 7 = -3(x – 11) passes through.
Solution:
Given equation of a line is:
y – 7 = -3(x – 11)
Comparing the above equation with y – y1 = m(x – x1)
m = -3
(x1, y1) = (11, 7)
Therefore, slope is -3 and the point is (11, 7).
Applications
- Used to find the equation of a line when a single point and the slope are given.
- Helps in solving problems related to tangents and normals in coordinate geometry.
- Used in physics to represent motion equations in graphs.
Conclusion
The point-slope form is an essential formula in coordinate geometry. It simplifies calculations and is useful in various applications like physics, engineering, and economics where linear relationships are involved.
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📕 Written by: Neeraj Anand
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