Two-point form

In geometry, a line is a collection of points that can be stretched in two directions indefinitely. To put it another way, a line is formed by continually extending the two endpoints in any direction.

The equation of a line in the coordinate plane can be expressed in two-point form. The equation of a line can be found using a variety of ways, depending on the information provided. One of the ways is the two-point form. When two points on a line are supplied, this is used to get the equation of the line. Slope intercept form, intercept form, point-slope form, and so on are some more useful ways to write a line equation.

Two-point form:-

One of the most common ways to describe a straight line algebraically is in two-point form. Each point on the line is represented by the equation of a line, which is satisfied by each point on the line. The two-point form of a line is used to find a line’s equation given two points on it (x1, y1 ) and (x2, y2 ).

Line Equation in Two-Point Form:-

A line going through these two points has the two-point form:

y-y1 =[(y2-y1)/(x2-x1]×(x-x1)

Or,

y-y2=[(y2-y1)/(x2-x1)]×(x-x2)

We keep ‘x’ and ‘y’ as variables, and (x, y) represents any random point on the line.

Formula For Two-Point Form:-

The two-point formula is used to algebraically represent a line using the coordinates of two points on the line. 

Let (x1, y1) and (x2, y2) be two points such that the equation of a line passes between them. The two-point form formula is as follows:

Two-point form formula:-

y-y1 =[(y2-y1)/(x2-x1)]×(x-x1)

Or,

y-y2=[(y2-y1)/(x2-x1]×(x-x2)

where,

An arbitrary point on the line is (x, y).

The coordinates of points lying on the line are (x1, y1) and (x2, y2 ).

The formula for Two-Point Form Derivation:-

Given two points on a line, we can obtain the two-point form equation for that line. Consider two fixed points on a line in a coordinate plane, P(x1, y1 ) and Q(x2, y2 ). Assume that R(x, y) is any point on the line at random.

Because P, Q, and R are all on the same axis:

PR’s slope is equal to PQ’s slope.

To calculate the slope, use the slope formula,

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

(x – x1 ) is multiplied on both sides.

So,

y-y1 =[(y2-y1)/(x2-x1)]×(x-x1)

The two-point form was derived. It’s used to get the equation of a line passing through two locations.

In a similar way, we may obtain another two-point form formula, 

y-y2=[(y2-y1)/(x2-x1]×(x-x2)

Using the two-point form, get the equation of a line:-

As previously mentioned, the equation of a line can be found by using two points on the line. To find the straight-line equation, we can use the two-point form and follow the methods outlined below.

Step 1: Write down (x1,y1) and (x2,y2) as the coordinates of the two points on the line.

Step 2: Use the two-point formula,y-y1 =[(y2-y1)/(x2-x1)]×(x-x1)

to solve the problem.

Step 3: To represent the line, simplify the resulting equation to the form y = mx + b.

Important Notes on Two-Point Form:

• The two-point form of a line can also be expressed as: 

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

Or,

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

• (x = a) is the equation for a vertical line passing through (a, b).

     This is a one-of-a-kind situation in which the two-point form can’t be used.

Conclusion:-

The two-point form of a line travelling through the points (x1,y1) and (x2,y2) in the Cartesian plane is given by

y-y1 =[(y2-y1)/(x2-x1)]×(x-x1)

, or equivalently, 

y-y2=[(y2-y1)/(x2-x1]×(x-x2).

The equation of a straight line in two-point form, or the equation of a straight line passing through two points.

A line going through two points (x1,y1) and (x2,y2) has the equation

y-y1 =[(y2-y1)/(x2-x1)]×(x-x1).