# 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 + $$2at_1t_2$$.