Two Point Form – Solved Examples

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

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.