Equation of Horizontal Line

Theorem
Let $\LL$ be a horizontal line embedded in the Cartesian plane $\CC$.

Then the equation of $\LL$ can be given by:
 * $y = b$

where $\tuple {0, b}$ is the point at which $\LL$ intersects the $y$-axis.


 * Graph-of-horizontal-line.png

Proof
From the Normal Form of Equation of Straight Line in Plane, a general straight line can be expressed in the form:


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

where:
 * $p$ is the length of a perpendicular $\PP$ from $\LL$ to the origin
 * $\alpha$ is the angle made between $\PP$ and the $x$-axis.

As $\LL$ is horizontal, then by definition $\PP$ is vertical.

By definition, the vertical line through the origin is the $y$-axis itself.

Thus:
 * $\alpha$ is a right angle, that is $\alpha = \dfrac \pi 2 = 90 \degrees$
 * $p = b$

Hence the equation of $\LL$ becomes:

Hence the result.

Also see

 * Equation of Vertical Line