Straight line in Two-point Form

We will learn how to find the equation of a straight line in two-point form or the equation of the straight line through two given points.

The equation of a line passing through two points (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) is y - y$$_{1}$$ = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$(x - x1)

Let the two given points be (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$).

We have to find the equation of the straight line joining the above two points.

Let the given points be A (x$$_{1}$$, y$$_{1}$$), B (x$$_{2}$$, y$$_{2}$$) and P (x, y) be any point on the straight line joining the points A and B.

Now, the slope of the line AB is $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$

And the slope of the line AP is $$\frac{y - y_{1}}{x - x_{1}}$$

But the three points A, B and P are collinear.

Therefore, slope of the line AP = slope of the line AB

⇒ $$\frac{y - y_{1}}{x - x_{1}}$$ = $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$

⇒ y - y$$_{1}$$ = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$ (x - x$$_{1}$$)

The above equation is satisfied by the co-ordinates of any point P lying on the line AB and hence, represents the equation of the straight line AB.

Solved examples to find the equation of a straight line in two-point form:

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

Solution:

The equation of the straight line passing through the points (2, 3) and (6, - 5) is

$$\frac{ y - 3}{ x + 2}$$ =  $$\frac{3 + 5}{2 - 6}$$,[Using the form,  $$\frac{y - y_{1}}{x - x_{1}}$$ = $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$]

⇒ $$\frac{ y - 3}{ x + 2}$$ = $$\frac{ 8}{ -4}$$

⇒ $$\frac{ y - 3}{ x + 2}$$ = -2

⇒ y - 3 = -2x - 4

⇒ 2x + y + 1 = 0, which is the required equation

2. Find the equation of the straight line joining the points (- 3, 4) and (5, - 2).

Solution:

Here the given two points are (x$$_{1}$$, y$$_{1}$$) = (- 3, 4) and (x$$_{2}$$, y$$_{2}$$) = (5, - 2).

The equation of a line passing through two points (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) is y - y$$_{1}$$ = [$$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$](x - x$$_{1}$$).

So the equation of the straight line in two point form is

y - y$$_{1}$$ = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$ (x - x$$_{1}$$)

⇒ y - 4 = $$\frac{-2 - 4}{5 - (-3)}$$[x - (-3)]

⇒ y - 4 = $$\frac{ -6}{ 8}$$(x + 3)

⇒ y - 4 = $$\frac{ -3}{ 4}$$(x + 3)

⇒ 4(y - 4) = -3(x + 3)

⇒ 4y - 16 = -3x - 9

⇒ 3x + 4y - 7 = 0, which is the required equation.