Equation of Straight Line in Plane/Two-Point Form/Parametric Form

Theorem

Let $\LL$ be a straight line embedded in a cartesian plane, given in two-point form as:

$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$

Then $\LL$ can be expressed by the parametric equations:

$\begin {cases} x = x_1 + t \paren {x_2 - x_1} \\ y = y_1 + t \paren {y_2 - y_1} \end {cases}$

Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.

Let $t = \dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$.

We then have:

$\ds t$

$=$

$\ds \dfrac {x - x_1} {x_2 - x_1}$

$\ds \leadsto \ \ $

$\ds x - x_1$

$=$

$\ds t \paren {x_2 - x_1}$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds x_1 + t \paren {x_2 - x_1}$

$\ds t$

$=$

$\ds \dfrac {y - y_1} {y_2 - y_1}$

$\ds \leadsto \ \ $

$\ds y - y_1$

$=$

$\ds t \paren {y_2 - y_1}$

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds y_1 + t \paren {y_2 - y_1}$

The result follows.

$\blacksquare$

1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $4$. Special forms of the equation of a straight line: $(3)$ Line through two points