"Two point form equation of a line" is one of the ways to express the equation of a line.

Here, we will form equation of a straight line using the given two points.

**Two point form equation of a line**

Here, the two points are

**(x₁ , y₁) and (x₂ , y₂)**

**Problem 1 :**

Find the general equation of the straight line passing through the points (-1, 1) and (2, -4).

**Solution :**

Given : Two points on the straight line are (-1, 1) and (2, -4)

So, the equation of the straight line in two-points form is

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

Plugging (x₁ , y₁) = (-1, 1) and (x₂, y₂) = (2, -4), we get

(y - 1) / (-4 - 1) = (x + 1) / (2 + 1)

(y - 1) / (-5) = (x + 1) / 3 ----------> 3(y - 1) = -5(x + 1)

3y - 3 = -5x - 5 ----------> 5x + 3y + 2 = 0

**Hence the general equation of straight line is 5x + 3y + 2 = 0**

**Problem 2 :**

Find the general equation of the straight line passing through the points (-2, 5) and (3, 6).

**Solution :**

Given : Two points on the straight line are (-2, 5) and (3, 6)

So, the equation of the straight line in two-points form is

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

Plugging (x₁ , y₁) = (-2, 5) and (x₂, y₂) = (3, 6), we get

(y - 5) / (6 - 5) = (x + 2) / (3 + 2)

(y - 5) / 1 = (x + 2) / 5 ----------> 5(y - 5) = (x + 2)

5y - 25 = x + 2 ----------> x - 5y + 27 = 0

**Hence the general equation of straight line is x - 5y + 27 = 0**

**Problem 3 :**

The vertices of a triangle ABC are A(2, 1), B(-2, 3) and C(4, 5). Find the equation of the median through the vertex A.

**Solution :**

Median is a straight line joining a vertex and the midpoint of the opposite side.

Let D be the midpoint of BC.

The median through A is nothing but the line joining two points A (2,1) and D(1, 4).

So, equation of the median through A is

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

Plugging (x₁ , y₁) = (2, 1) and (x₂, y₂) = (1, 4), we get

(y - 1) / (4 - 1) = (x - 2) / (1 - 2)

(y - 1) / 3 = (x - 2) / (-1) ----------> -1(y - 1) = 3(x - 1)

- y + 1 = 3x - 3 ----------> 3x + y - 4 = 0

**Hence, the equation of the median through A is ****3x + y - 4 = 0. **

Apart from two point form equation of a straight line, we have some other different forms of equation of a straight line.

They are

(i) Slope intercept form

(ii) Point slope form

(iii) Intercept form.

Let us look at the above different forms of equation of a straight line in detail.

Here,

**Slope of the line = m **

**y-intercept = b **

Here,

**Slope of the line = m **

**Point = (x₁ , y₁)**

Here,

**x- intercept = a**

**y- intercept = b**

Apart from the above forms of equation of straight line, there are some other ways to get equation of a straight line.

1. If a straight line is passing through a point (0,k) on y-axis and parallel to x-axis, then the equation of the straight line is y = k

2. If a straight line is passing through a point (c,0) on x-axis and parallel to y-axis, then the equation of the straight line is x = c

3. Equation of x-axis is y = 0.

(Because, the value of "y" in all the points on x-axis is zero)

4. Equation of y-axis is x = 0.

(Because, the value of "x" in all the points on y-axis is zero)

5. General equation of a straight line is

ax + by + c = 0

After having gone through above stuff, we hope that students would have understood the stuff "Two point form equation of a line".

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