Equation of Straight Line in Plane

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Theorem

General Equation

A straight line $\mathcal L$ is the set of all $\tuple {x, y} \in \R^2$, where:

$\alpha_1 x + \alpha_2 y = \beta$

where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.


Slope-Intercept Form

Let $\mathcal L$ be the straight line defined by the general equation:

$\alpha_1 x + \alpha_2 y = \beta$


Then $\mathcal L$ can be described by the equation:

$y = m x + c$

where:

\(\displaystyle m\) \(=\) \(\displaystyle -\dfrac {\alpha_1} {\alpha_2}\)
\(\displaystyle c\) \(=\) \(\displaystyle \dfrac {\beta} {\alpha_2}\)

such that $m$ is the slope of $\mathcal L$ and $c$ is the $y$-intercept.


Two-Intercept Form

Let $\mathcal L$ 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 $\mathcal L$ can be described by the equation:

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


Normal Form

Let $\mathcal L$ be a straight line such that:

the perpendicular distance from $\mathcal L$ to the origin is $p$
the angle made between that perpendicular and the $x$-axis is $\alpha$.


Then $\mathcal L$ can be defined by the equation:

$x \cos \alpha + y \sin \alpha = p$


Two-Point Form

Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.

Let $\mathcal L$ be the straight line passing through $p_1$ and $p_2$.


Then $\mathcal L$ can be described by the equation:

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

or:

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


Point-Slope Form

Let $\mathcal L$ be a straight line embedded in a cartesian plane, given in slope-intercept form as:

$y = m x + c$

Let $\mathcal L$ pass through the point $\tuple {x_0, y_0}$.


Then $\mathcal L$ can be expressed by the equation:

$y - y_0 = m \paren {x - x_0}$