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 + \(2at_1t_2\).

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