Point Slope Form Formula: Equation of Straight Line & Solved Questions

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Point Slope Form is a formula that is used to find the equation of a line. In mathematics, the point slope form is used to calculate the equation for a straight line which shows more affinity towards the x axis and passes through a given point. We use this formula because we can easily calculate the equation of any given line with the help of slope and a pair of coordinate points.

Table of Content

What is Point Slope Form?
What is Slope?
Point Slope Form Formula
Applications in Real Life
Things to Remember
Sample Questions

Key Terms: Point Slope Form, Slope, Equation of a Straight Line, Coordinates, Two-point Slope Form, Straight Line

What is Point Slope Form?

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In geometry, the equation of a straight line, present on a coordinate plane, may be represented in various ways. Some of these forms are:

Standard form
Horizontal form
Point slope form
Vertical form

It should be kept in mind that in a slope, there may be different lines present. But when we have certain information regarding its activity, for example, which point it passes through, we can easily deduce it to get a proper equation of that straight line. The point slope form formula is a way to represent the point-slope of the equation of a given straight line. It uses the slope of the line, and the point through which the line passes to measure the point-slope.

Point-Slope Form

What is Slope?

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Slope or the gradient is used often in reference to a line. It is a number that indicates the steepness or rise, and the direction of the line. Earlier, it has been derived from the formula y = mx + c, where ‘m’ is the slope. It calculates the vertical change to the horizontal change. The rise of a slope is calculated by absolute value. This means, the higher the absolute value, the higher is the slope.

Point Slope Form Formula

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Slope is denoted as the difference between the y coordinates divided by the difference in x coordinates.

Slope = difference in the y coordinates/ difference in the x coordinates

Therefore, for the given coordinates P (x,y) and Q (a,b), the slope happens to be,

m = (y - b)/(x - a)

m(x – a) = y – b

y – b = m(x – a)

The points (x, y) can be referred to as the points which lie on the line and satisfy the given equation.

In any given case, where there are two different sets of coordinates involved, the formula can be extended and modified. In this case, let the sets be (x1, y1) and (x2, y2). This becomes a two-point slope form.

Here, the formula for slope becomes,

m = (y₂ – y₁) / (x₂ – x₁)

If we substitute m in the one-point slope form, by taking the points (x1, y1), we get

y – y₁= (y₂ – y₁) / (x₂ – x₁) (x – x₁)

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Applications in Real Life

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The applications of the point-slope form formula are as follows:

The point-slope formula serves as an important tool in calculus. You can use it to find the slope taking the given derivative into consideration.
It also helps in solving problems in algebra.

Things to Remember

The point slope form formula is used to define the point-slope of the equation of a given line. It does so with the help of the slope m, x coordinate, and y coordinate.
It can be said that a linear equation in two variables represents a given line.
The slope of a line is referred to as a number that gives information regarding the direction and the level of rise of a line. It is also known by the term gradient.
We can only use the point-slope formula when we have the slope of the line, along with a pair of coordinates or points. The formula for a point-slope form is given as (y- y₁) = m(x - x₁)
The equation of a vertical line passing through (a, b) is of the form x = a. On the other hand, in the case of a horizontal line, it is of the form y = b.

Sample Questions

Ques. What will be the equation of the line which passes through the coordinates (4, 8)? The given slope is 7. (3 Marks)

Ans. Given that the slope of the line, m = 7

(x₁, y₁) = (4,8)

According to the point-slope formula,

y - y₁ = m(x - x₁)

y - 8 = 7(x - 4)

y - 8 = 7x - 28

y - 8 + 28 = 7x

y + 20 = 7x

y - 7x + 20 = 0

Therefore, the equation of the line is y - 7x + 20 = 0.

Ques. The equation of a given line is 3x - 7y + 1. Calculate the slope. (2 Marks)

Ans. According to the formula, we know that,

y - y1 = m (x - x1)

Therefore, x1 = 3, and y1 = -7

Slope of the line = -(3/-7)

Hence, the slope of the given line is 3/7.

Ques. Calculate the equation of the line whose slope is 5, and the coordinate points are (3, 7) respectively. (3 Marks)

Ans. Given that, m = 5

(a, b) = (3, 7)

According to the point-slope formula,

y-b = m (x-a)

y - 7 = 5 (x - 3)

y - 7 = 5x - 15

y - 7 + 15 = 5x

y - 8 = 5x

5x - y + 8 = 0

So, the equation of the line is 5x - y + 8 = 0.

Ques. Where is the point-slope form formula used? (2 Marks)

Ans. The point-slope formula is mainly used in calculus. It comes to use after you have found out the slope using the derivative. The formula is used extensively in many algebraic operations too and thereby helps solve problems in the real world.

Ques. What will be the equation of the line which passes through two points (6, 8) and (4, 10)? (5 Marks)

Ans. We have been provided with 2 separate sets of coordinate points. We have,

x1 = 6, x2 = 4, y1 = 8, and y2 = 10.

According to the two-point slope formula,

y - y₁ = (y₂ - y₁) / (x₂ - x₁) (x - x₁)

y - 12 = (10 - 8) / (4 - 6) (x - 6)

y - 12 = 2/-2 (x - 6)

y - 12 = -1 (x - 6)

y - 12 = -x + 6

y - 12 - 6 + x = 0

y - 18 + x = 0

The equation of the line is y - 18 + x= 0.

Ques. A given line crosses two points on its way. These are given as (2, 4) and (4, 6). What is the equation of the line? (5 Marks)

Ans. According to the question,

x1 = 2, x₂ = 4, y1 = 4, y₂ = 6

The 2-point slope formula lays down that whenever there are two pairs of coordinates involved

y-y₁ = (y₂ - y₁)/(x₂ - x₁) (x - x₁)

y - 4 = (6-4) / (4-2)(x-2)

y - 4 = 2 / 2 (x-2)

y - 4 = 1 (x-2)

y - 4 = x - 2

y - 4 + 2 = x

y - 2 - x = 0

The equation of the line is y - 2 - x = 0.

Ques. What is a slope? (2 Marks)

Ans. The slope or gradient is a number that represents the steepness of a line. It also throws light on its direction and how it is moving. It is denoted by the letter ‘m’ and is calculated by absolute value. The higher the value, the steeper it is. The slope of any given line is calculated by taking the difference between the y coordinates to the difference in the x coordinates.

Slope = difference in the y coordinates/difference in the x coordinates

Ques. Mention some features of the slope of a line. (3 Marks)

Ans. Some important characteristics of a slope are as follows:

If the line moves from left to right and appears to be moving in an upward direction, the slope is called positive. Here, m>0.
If the line is moving from left to right but in a downward direction, it is called a negative slope. Here, m is less than 0.
When the line is horizontal, the slope automatically becomes zero.
In the case of a vertical line, slope becomes undefined.

Ques. Coordinate points of a line are given as (5, 10). The slope of the line is 2. What equation will be formed? (3 Marks)

Ans. According to the point-slope formula,

y - b = m (x-a)

Slope = 2

(a,b) = (5, 10)

y - y₁ = m (x - x₁)

y - 10 = 2 (x -5)

y - 10 = 2x - 10

y - 10 + 10 - 2x = 0

y - 2x = 0

Hence, the equation of the given line is y - 2x = 0.

Ques. Find the slope of a line with the equation y - 14 = -6 (x - 22). (3 Marks)

Ans. Given that, y - 14 = -6 (x - 22)

As we can clearly see the above-given equation is in the form of y – y1 = m (x – x1), it can be concluded that the slope of the line, m = -6.

Also, x₁ = 22, and y₁ = 14

Therefore, coordinates are (x₁, y₁) = (22, 14)

Ques. The points of a given line are (10, 20). The slope of this line is 5. What will be the equation? (3 Marks)

Ans. We know that m = 5.

Given that (x₁, y₁) = (10, 20)

Substituting in the formula,

y - y₁ = m (x - x₁)

y - 20 = 5 (x - 10)

y - 20 = 5x - 50

y - 20 + 50 = 5x

y + 30 = 5x

y - 5x + 30 = 0

Therefore, the equation of the line is y - 5x + 30 = 0.

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Point Slope Form Formula: Equation of Straight Line & Solved Questions

What is Point Slope Form?

What is Slope?

Point Slope Form Formula

Applications in Real Life

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find the following integral: \(\int (ax^2+bx+c)dx\)

2.
Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

3.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

4.
For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?

5.
By using the properties of definite integrals, evaluate the integral: \(∫_0^π log(1+cosx)dx\)

6.
Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

Similar Mathematics Concepts

Comments

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Point Slope Form Formula: Equation of Straight Line & Solved Questions

What is Point Slope Form?

What is Slope?

Point Slope Form Formula

Applications in Real Life

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. Find the following integral: \(\int (ax^2+bx+c)dx\)

2. Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

3. Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

4. For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?

5. By using the properties of definite integrals, evaluate the integral: \(∫_0^π log(1+cosx)dx\)

6. Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find the following integral: \(\int (ax^2+bx+c)dx\)

2.
Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

3.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

4.
For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?

5.
By using the properties of definite integrals, evaluate the integral: \(∫_0^π log(1+cosx)dx\)

6.
Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)