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:

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

Proof

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:

\(\ds \triangle OBP + \triangle OAP\)

\(=\)

\(\ds \triangle OAB\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {b x} 2 + \dfrac {a y} 2\)

\(=\)

\(\ds \dfrac {a b} 2\)

Area of Triangle in Terms of Side and Altitude

\(\ds \leadsto \ \ \)

\(\ds b x + a y\)

\(=\)

\(\ds a b\)

multiplying both sides by $2$

\(\ds \leadsto \ \ \)

\(\ds \dfrac x a + \dfrac y b\)

\(=\)

\(\ds 1\)

dividing both sides by $a b$ and rearranging

$\blacksquare$

Sources

• 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