Equation of Horizontal Line

Theorem

Let $\mathcal L$ be a horizontal line embedded in the Cartesian plane $\mathcal C$.

Then the equation of $\mathcal L$ can be given by:

$y = b$

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

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 $\mathcal P$ from $\mathcal L$ to the origin
$\alpha$ is the angle made between $\mathcal P$ and the $x$-axis.

As $\mathcal L$ is horizontal, then by definition $\mathcal P$ 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 $\mathcal L$ becomes:

 $\displaystyle x \cos \dfrac \pi 2 + y \sin \dfrac \pi 2$ $=$ $\displaystyle b$ $\displaystyle \leadsto \ \$ $\displaystyle x \times 0 + y \times 1$ $=$ $\displaystyle b$ Sine of Right Angle, Cosine of Right Angle $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle b$

Hence the result.

$\blacksquare$