Equation of Straight Line in Plane/Two-Intercept Form/Proof 3

Theorem

Let $\LL$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.

Then $\LL$ can be described by the equation:

$\dfrac x a + \dfrac y b = 1$

We have that $\LL$ is passes through the two points $A = \tuple {a, 0}$ and $B = \tuple {0, b}$.

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

We have:

$\ds \triangle OAP + \triangle OPB$

$=$

$\ds \triangle OAB$

$\ds \leadsto \ \ $

$\ds a y + b x$

$=$

$\ds a b$

$\ds \leadsto \ \ $

$\ds \dfrac x a + \dfrac y b$

$=$

$\ds 1$

$\blacksquare$

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: $(2)$ Intercept form