Equation of Straight Line in Plane/Two-Intercept Form
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 1
From the General Equation of Straight Line in Plane, $\LL$ can be expressed in the form:
- $(1): \quad \alpha_1 x + \alpha_2 y = \beta$
where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.
Substituting for the two points whose coordinates we know about:
| \(\ds x = a, y = 0: \, \) | \(\ds \alpha_1 \times a + \alpha_2 \times 0\) | \(=\) | \(\ds \beta\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \alpha_1\) | \(=\) | \(\ds \dfrac \beta a\) |
and:
| \(\ds x = 0, y = b: \, \) | \(\ds \alpha_1 \times 0 + \alpha_2 \times b\) | \(=\) | \(\ds \beta\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \alpha_2\) | \(=\) | \(\ds \dfrac \beta b\) |
We know that $\beta \ne 0$ because none of $a, b, \alpha_1, \alpha_2$ are equal to $0$.
Hence:
| \(\ds \dfrac \beta a x + \dfrac \beta b y\) | \(=\) | \(\ds \beta\) | substituting for $\alpha_1$ and $\alpha_2$ in $(1)$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac x a + \dfrac y b\) | \(=\) | \(\ds 1\) | dividing both sides by $\beta$ |
$\blacksquare$
Proof 2
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:
| \(\ds \dfrac {y - 0} {x - a}\) | \(=\) | \(\ds \dfrac {b - 0} {0 - a}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -a y\) | \(=\) | \(\ds b x - b a\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac x a + \dfrac y b\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Proof 3
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:
| \(\ds \triangle OAP + \triangle OPB\) | \(=\) | \(\ds \triangle OAB\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a y + b x\) | \(=\) | \(\ds a b\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac x a + \dfrac y b\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Also known as
The two-intercept form of the Equation of Straight Line in Plane is also known as the intercept form.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: $10.5$: Equation of Line in terms of $x$ Intercept $a \ne 0$ and $y$ Intercept $b \ne 0$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): line: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): line: 2.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): line (in two dimensions) Two-intercept form
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): straight line (in the plane)