Two Point Form – Definition, Formula & Derivation

Two-point form of a line is the equation of a line when two points on a line are given, the two-point form formula is Y − y₁ = (y₂ − y₁)/(x₂ − x₁)(X − x₁). Where the two points are, (x₁, y₁) and (x₂, y₂). If in geometry two points are given then the equation of the line passing through these two points is given using the two-point form of the line.

In this article, we will learn about the two-point form, the equation of a line in the two-point form, Two Point Form examples, derivation of the two-point form, and others in detail.

Two-point form in math is an important way of finding the equation of a line when the coordinates of any two points on the line are known. The equation of a line represents every point on the line, and we can say that it is satisfied by each point on the line.

Two-point form is one of the various methods used to find out the equation of the line. We use this method to calculate the line equation when we know two points’ coordinates on the same line. Therefore, the name comes two-point form of line.

Let A(x₁, y₁) and B(x₂, y₂) be two co-ordinates of the two distinct points given on the line as shown in the image added below:

Then, the formula of 2 point form of the equation is:

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

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

where,

• x and y are arbitrary points on the line.

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

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

where,

• (x, y) is an arbitrary point on the line.
• (x₁, y₁) and (x₂, y₂) are coordinates of points lying on the line.

Let M(x₁, y₁) and N(x₂, y₂) be the two given points on the line L and let P(x,y) be a random point on the line L

From the figure, we can observe that the three points M , N and P lie on the same line. Hence, they are collinear.

Slope of line MP = Slope of line NP

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

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

Equation of line in two point form is found using the steps added below as:

Step 1: First, take any two points that lie on the line. Let’s these points be (x₁, y₁) and (x₂, y₂).

Step 2: Use the two-point line formula to find the equation of the line as:

• (y – y₁) = (y₂ – y₁)/(x₂ – x₁){x – x₁}

Step 3: Simplify the equation to get your required equation.

This is explained using the example added below:

Example: Find equation of a line passing through the points (1,2) and (3,4)?

Given points

• A = (1, 2)
• B = (3, 4)

Therefore, x₁ = 1, y₁ = 2, x₂ = 3, y₂ = 4

Equation of line in two point form,

• (y – y₁) = (y₂ – y₁)/(x₂ – x₁){x – x₁}

Substitute the values

(y – 4) = (2 – 4)(x-3)/(1 – 3)

(y – 4) = (-2)(x – 3)/(-2)

y – 4 = x – 3

y = x -3 + 4

y = x + 1

Thus, the equation of the line is: y = x + 1

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Example 1: Find the equation of the line passing through the points A(-2, 3) and B(3, 5).

Solution:

Given points are:

• A = (-2, 3)
• B = (3, 5)

Using the formula, we get:

⇒ (y-3) = {(5 – 3)/(3 -(-2))}.(x + 2)

⇒ (y – 3) = 2/5.(x+2)

⇒ 5y – 15 = 2x + 4

⇒ 5y = 2x + 19

Thus, the equation of the line is 5y = 2x + 19

Example 2: Find the equation of the line passing through the points A(0,3) and B(3,0).

Solution:

Given points are:

• A = (0, 3)
• B = (3, 0)

Using the formula, we get:

⇒(y – 0) = {(3 – 0)/(0 – (-3)}(x-3)

⇒ y = {3/3}(x-3)

⇒ 3y = 3x-9

Thus, equation of the line is 3y = 3x – 9

Example 3: Find the equation of a straight line whose x-intercept is ‘a’ and y-intercept is ‘b’ ?

Solution :

Given points are:

• A = (a, 0)
• B = (0, b)

Using the formula, we get:

⇒ (y-0) = (b-0) (x-a) / (0-a)

⇒ y = b(x-a) / (-a)

⇒ -ay = bx – ba

⇒ ay + bx = ab

Thus, the equation of the line is ay + bx = ab

Example 4: Write the equation of the line through the points (3, –3) and (1, 5).

Solution:

Given points are:

• A = (3, -3)
• B = (1, 5)

Using the formula, we get:

⇒ (y + 3) = (5 + 3) (x – 3) / (1-3)

⇒ (y + 3) = -4(x – 3)

⇒ y+3 = -4x+12

⇒ 4x + y = 9

Thus, the equation of the line is 4x + y = 9

Example 5: Derive the y-intercept of the line with the coordinates given by A(3,-2) & B(1,-3) passing through it and also find the slope m of the line.

Solution:

Given points are:

• A = (3, -2)
• B = (1, 5)

Using the formula, we get:

⇒ (y + 2) = (5 + 2) (x – 3) / (1-3)

⇒ (-2)(y + 2) = 7(x – 3)

⇒ -2y – 4 = 7x – 21

⇒ 7x + 2y = 17

Thus, the equation of the line is 7x + y = 9

To find slope compare the given equation with,

y = mx + c

Given equation:

7x + y = 9

⇒ y = -7x + 9

Hence, m = -7

Thus, the slope of the line is -7

Important Maths Related Links:

• Euclidean Distance Formula
• x and y axis
• Geometry

Q1: What is the slope of a line passing through the points (5, -4) and (-3, 6)?

Q2: What is the slope of a line passing through the points (5, 0) and (0, 5)?

Q3: Derive the y-intercept of the line with the coordinates given by A(3, -2) and B(-1, 3) passing through it and also find the slope m of the line.

Q4: What is the equation of a vertical line passing through the point A(4, -7).

Q5: What is the slope of a line passing through the points (10, 5) and (6, 12)?

Q6: What is the slope of a line passing through the points (3, -9) and (-3, -7)?

What is the two point form?

Two-point form of a line is the equation of a line when two points on a line are given, the two-point form formula is Y − y₁ = (y₂ − y₁)/(x₂ − x₁)(X − x₁). Where the two points are, (x₁, y₁) and (x₂, y₂)

How do you determine whether a point lies on a line?

Every point on a line satisfies its line equation. For example, to see whether (3, 6) lies on a line y=2x, we substitute x=3 & y=6 in the given equn. Then we get – 6=2(3) or 6=6. The equation is satisfied and hence the point (3, 6) lies on the line y=2x.

What is the point slope form?

The equation of line in point slope form is given by : (y-y₁) = m(x-x₂) .

What is the equation of x-axis?

The equation of x-axis is : y=0

What is the equation of y-axis?

The equation of y-axis is : x=0

Give an example of a two point form?

Point A with coordinates (2, 3) and Point B with coordinates (4, 7). To find the equation of the line that goes through these points, we substitute the coordinates into the two-point formula:

• x₁ = 2, y₁ ​= 3
• x₂ = 4, y₂ = 7

Substituting these values into the equation gives us:

y−3=(7−3)/(4−2)