Here you will learn two point form of a line equation with proof and examples.
Letโs begin โ
Two Point Form of a Line
The equation of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) isย
\(y โ y_1\) = (\(y_2 โ y_1\over x_2 โ x_1\))(\(x_2 โ x_1\))
Proof :ย
Let m be the slope of line passing through \((x_1, y_1)\) and \((x_2, y_2)\). Then,
m = \(y_2 โ y_1\over x_2 โ x_1\)
By using point-slope form, the equation of the line is,
\(y โ y_1\) = m(\(x_2 โ x_1\))ย ย ย ย ย ย ย ย ย ย ย ย
\(y โ y_1\) = (\(y_2 โ y_1\over x_2 โ x_1\))(\(x_2 โ x_1\))
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
\(y โ y_1\) = (\(y_2 โ y_1\over x_2 โ x_1\))(\(x_2 โ x_1\))
\(\implies\) y โ 3 = \(3 โ (-2)\over -1 โ 4\)(x + 1)
\(\implies\) y โ 3 = -x โ 1 \(\implies\) x + y โ 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
\(y โ y_1\) = (\(y_2 โ y_1\over x_2 โ x_1\))(\(x_2 โ x_1\))
y โ \(2at_1\) = \(2at_2 โ 2at_1\over {at_2}^2 โ {at_1}^2\) \((x โ a{t_1}^2)\)
y โ \(2at_1\) = \(2\over t_1 + t_2\) \((x โ a{t_1}^2)\)
\(\implies\) y\((t_1 + t_2)\) โ \(2a{t_1}^2\) โ \(2at_1t_2\) = 2x โ \(2a{t_1}^2\)
\(\implies\) y\((t_1 + t_2)\) = 2x + \(2at_1t_2\).
