The two-point form is a way to write the equation of a straight line when you know two points that lie on that line.

If you are given two points, say (x₁, y₁) and (x₂, y₂), you can use this form to find the equation of the line passing through both points.

This equation shows all the points that lie exactly on the line. In other words, every point on the line will satisfy the equation.

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The two-point form is very helpful when you don’t know the slope directly, but you do know two points on the line.

In geometry, a line can be defined as the shortest distance between any two points or distance between two lines. The slope of the line tells us how steeply the line rises or falls. The slope-intercept form of a line is given by y = mx + b where m is the slope and b is the y-intercept of the line. In this article, we will learn about the two-point form, formula, General Equation of a Line in Two Point Form and its derivation with solved examples.

Two point formula is a way of expressing the equation of a line given any two points through which the line passes. We represent the equation of a straight line using the formula:

 ax + by = c, where x and y are variables.

The two-point form of a line is used to find the equation of a line given two points (x1,y1),(x2,y2).

A straight line has 2 very important properties:

1. Slope of a line
    
2. Y–intercept of the line

Slope of a Line

A minimum of two points make up a unique line. The rise and the run are terms used to describe changes in height between two points. The rise divided by the run gives you the slope of a line. Using the general line equation y = mx + b, the slope, m, may be determined.

The formula for the slope of the line is given by:
m = rise / run = (y₂ − y₁) / (x₂ − x₁)

m – slope

(x1,y1) – coordinates of first point in the line

(x2,y2) – coordinates of second point in the line

Y−intercept of the line

The location on a graph where the line crosses the x-axis and y-axis is known as the y-intercept. A function must pass the vertical lines test, or have a y-intercept, in order to qualify as an equation of the line. A line can only have one y-intercept at most.

Two Point Form of the Equation of a Line 

Let’s say you have two points on a line, called P₁ (x₁, y₁) and P₂ (x₂, y₂).
Now, imagine another point P(x, y) that also lies on the same line.

Since all three points (P₁, P₂, and P) are on the same straight line, they are collinear.
That means the slope (or slant) between P₁ and P is the same as the slope between P₁ and P₂.

So we compare the change in y-values to the change in x-values between both sets of points.

This gives us the formula:

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

If we rearrange this equation, we get the two-point form of a straight line:

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

This is called the two-point form because it helps you write the equation of a line when you're given any two points on it.

Two Point Form Formula

Using the coordinates of two points that are on a line, the two-point formula is used to algebraically express the line.

Two point form formula for a line: \(y-y_1=\frac{(y_2-y_1)}{(x_2-x_1)}\times(x-x_1)\) or equivalently, \(y-y_2=\frac{(y_2-y_1)}{(x_2-x_1)} \times(x-x_2)\)

Equation of a Line in Two Point Form

A line going through these two points has a two-point form that is \(y-y_1=\frac{(y_2-y_1)}{(x_2-x_1)}\times(x-x_1)\) or equivalently, \(y-y_2=\frac{(y_2-y_1)}{(x_2-x_1)} \times(x-x_2)\). Here, the variables \(x\) and \(y\) represent any random point on the line represented by \((x, y)\).

Derivation of Two Point Form Formula

Two-point form of a line in the Cartesian plane passing through the points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(y-y_1=\frac{(y_2-y_1)}{(x_2-x_1)}\times(x-x_1)\) or equivalently, \(y-y_2=\frac{(y_2-y_1)}{(x_2-x_1)} \times(x-x_2)\)

Let’s derive the equation using the point-slope form of the equation.

Let’s assume the slope of the line to be m. Then, since the line passes through the point \((x_1, y_1)\), its equation will be:

\(y – y_1 = m(x – x_1)\) ……… (i)

However, we do not know the value of \(m\). Instead, we have been given another point \((x_2, y_2)\) , through which the line passes. This means that the coordinates \((x_2, y_2)\) must satisfy the above equation. We’ll get: \(y_2 – y_1 = m(x_2 – x_1)\)

\(m = \frac{(y_2 – y_1)}{(x_2 – x_1)}\)

This formula can give us the slope of the line. On substituting this in equation (i), we’ll get the required equation as \(y – y_1 = \frac{(y_2 – y_1)}{(x_2 – x_1)}\times(x – x_1)\). This also tells us one more thing. We can find the slope of the line joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \(\frac{(y_2 – y_1)}{(x_2 – x_1)}\)

Slope of the line AB equals \(tanθ\) or \(\frac{AC}{BC}\) or \(\frac{(y_2 – y_1)}{(x_2 – x_1)}\). 

Learn about Direction Ratios and Direction Cosines

Finding Equation of Line using Two Point Form

Steps to find the equation of a line using two point form:

• Step 1: Find the slope using the slope formula
  •  

Steps to find the equation of a line using two point form:

• Step 1: Find the slope using the slope formula
  • \({\text{Slope}}={\text{m}}=\frac{\text{rise}}{\text{run}}=\frac{{\text{y}}_2-{\text{y}}_1}{{\text{x}}_2-{\text{x}}_1}\)
  • \(\text{Point 1 or P}_{1}=(\text{x}_{1}, \text{y}_{1})\)
  • \(\text{Point 2 or P}_{2}=(\text{x}_{2}, \text{y}_{2})\)
• Step 2: Use the slope and one of the points to solve for the y-intercept (b). The \(x\) and \(y\) in your equation \(y = mx + b\) can be replaced by one of your points, and the \(m\) can be replaced by the slope you just calculated. Then \(b\) is the only variable left. Use the tools you know for solving for a variable to solve for \(b\).
• Step 3: Once you know the value for m and the value for \(b\), you can put these into the slope-intercept form of a line \((y = mx + b)\) to get the equation for the line.

Important Notes on Two-Point Form 

The two-point form is a way to write the equation of a straight line when you know two points on the line. The general formula looks like this:

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

This equation helps you find the line that passes through the two points (x₁, y₁) and (x₂, y₂). You can also use the second point instead, like this:

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

Both forms will give the same result. You can choose whichever is easier for you to use based on the point you're working with.

Special Case: If both points have the same x-value (for example, x₁ = x₂), then the line is vertical. The equation for a vertical line is simple:

x = a, where "a" is the constant x-value.

In this case, you cannot use the two-point form, because dividing by zero (x₂ − x₁ = 0) is not allowed in math.

So, always check if the x-values are different before using the two-point formula!.

Facts About Two-Point Form 

The two-point form is used to find the equation of a straight line when you know two points on the line. It can be written in two ways:

(y − y₁) / (x − x₁) = (y₂ − y₁) / (x₂ − x₁)
or
(y − y₂) / (x − x₂) = (y₂ − y₁) / (x₂ − x₁)

Both forms work the same way. You just choose which point to start with.
The two-point form cannot be used when both x-values are the same (that means x₁ = x₂). In this case, the line is vertical, and its slope is not defined.

For a vertical line going through a point like (a, b), the equation is simply:
x = a

This is because all points on a vertical line have the same x-value. Since the slope is undefined, the two-point formula doesn’t apply.

Two Point Form Examples

Now that we have learned various things about the Two Point Form. Let’s get some practice on the concepts through some examples.

Example 1: Find the general equation of a line passing through the points (-1, 1) and (3, -7).

Solution: The two points on the straight line are (-1, 1) and (3, -7).

Equation of a line in two-point form:

\(\frac{(y – y_1)}{(y_2 – y_1)} = \frac{(x – x_1)}{(x_2 – x_1)}\)

Substitute \((x_1, y_1) = (-1, 1)\) and \((x_2, y_2) = (3, -7)\).

\(\frac{(y – 1)}{(-7 – 1)} = \frac{(x + 1)}{(3 + 1)}\)

\(\frac{(y – 1)}{(-8)} =\frac{(x + 1)}{(4)}\)

\(4(y – 1) = -8(x + 1)\)

Distribute.

\(4y – 4 = -8x – 8\)

Simplify.

\(8x + 4y + 4 = 0\)

Divide each side by 2.

\(8x + 4y + 4 = 0\)

Example 2: Find the equation of the line joining the points (3, 6) and (2, -5).

Solution: \(x_1 = 3, y_1 = 6, x_2 = 2, y_2 = -5\)

Equation of line in two-point form:

\(\frac{(y – y_1)}{(y_2 – y_1)} = \frac{(x – x_1)}{(x_2 – x_1)}\)

Substitute \((x_1, y_1) = (3, 6)\) and \((x_2, y_2) = (6, -5)\).

\(\frac{(y – 6)}{-5 – 6)} = \frac{(x – 3)}{(2 – 3)}\)

\(\frac{(y – 6)}{(-11)} =\frac{(x – 3)}{(-1)}\)

\(-1(y – 6) = -11(x – 3)\)

\(1(y – 6) = 11(x – 3)\)

\(y – 6 = 11x – 33\)

\(11x – 33 – y + 6 = 0\)

Equation of the Line: \(11x – y – 27 = 0\)

Example 3: Find the equation of a line in a slope-intercept form which passing through the points (-2, 5) and (3, 6).

Solution: The two points on the straight line are (-2, 5) and (3, 6).

Equation of line in two-point form : \(\frac{(y – y_1)}{(y_2 – y_1)} = \frac{(x – x_1)}{(x_2 – x_1)}\)

Substitute \((x_1 , y_1) = (-2, 5)\) and \((x_2, y_2) = (3, 6)\).

\(\frac{(y – 5)}{(6 – 5)} = \frac{(x + 2)}{(3 + 2)}\)

\(\frac{(y – 1)}{1} = \frac{(x + 2)}{5}\)

\(5(y – 1) = x + 2\)

\(5y – 5 = x + 2\)

Add 5 to each side.

\(5y = x + 7\)

Divide each side by 5.

\(y = \frac{x}{5} + \frac{7}{5}\)

\(y = (\frac{1}{5})x + \frac{7}{5}\)

Hope this article on the Two Point Form was informative. Get some practice of the same on our free Testbook App. Download Now!. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: