Two Point Form of a Line Equation

Here you will learn two point form of a line equation with proof and examples.

Let’s begin –

Two Point Form of a Line

The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is 

$y – y_1$ = ($y_2 – y_1\over x_2 – x_1$)($x_2 – x_1$)

Proof : 

Let m be the slope of line passing through $(x_1, y_1)$ and $(x_2, y_2)$. Then,

m = $y_2 – y_1\over x_2 – x_1$

By using point-slope form, the equation of the line is,

$y – y_1$ = m($x_2 – x_1$)                       

$y – y_1$ = ($y_2 – y_1\over x_2 – x_1$)($x_2 – x_1$)

This is the required equation of the line.

Example : Find the equation of the line joining the points (-1, 3) and (4, -3).

Solution : Here, the two points are $(x_1, y_1)$ = (-1, 3) and $(x_2, y_2)$ = (4, -2).

So, the equation of the reuqired line is

$y – y_1$ = ($y_2 – y_1\over x_2 – x_1$)($x_2 – x_1$)

$\implies$ y – 3 = $3 – (-2)\over -1 – 4$(x + 1)

$\implies$ y – 3 = -x – 1 $\implies$ x + y – 2 = 0.

Example : Find the equation of the line joining the points $(a{t_1}^2, 2at_1)$ and $(a{t_2}^2, 2at_2)$.

Solution : Here, the two points are $(x_1, y_1)$ = $(a{t_1}^2, 2at_1)$ and $(x_2, y_2)$ = $(a{t_2}^2, 2at_2)$.

So, the equation of the required line is

$y – y_1$ = ($y_2 – y_1\over x_2 – x_1$)($x_2 – x_1$)

y – $2at_1$ = $2at_2 – 2at_1\over {at_2}^2 – {at_1}^2$ $(x – a{t_1}^2)$

y – $2at_1$ = $2\over t_1 + t_2$ $(x – a{t_1}^2)$

$\implies$ y$(t_1 + t_2)$ – $2a{t_1}^2$ – $2at_1t_2$ = 2x – $2a{t_1}^2$

$\implies$ y$(t_1 + t_2)$ = 2x + $2at_1t_2$.