Equation of Straight Line in Plane/Two-Intercept Form/Proof 2
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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
By definition, $\LL$ passes through $\tuple {a, 0}$ and $\tuple {0, b}$.
From the Equation of Straight Line in Plane through Two Points, $\LL$ can be expressed in the form:
\(\ds \dfrac {y - 0} {x - a}\) | \(=\) | \(\ds \dfrac {b - 0} {0 - a}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds -a y\) | \(=\) | \(\ds b x - b a\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac x a + \dfrac y b\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: Equation of a Straight Line: Intercept form