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.

● The Straight Line