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.

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:

\(\ds x \cos \dfrac \pi 2 + y \sin \dfrac \pi 2\)

\(=\)

\(\ds b\)

\(\ds \leadsto \ \ \)

\(\ds x \times 0 + y \times 1\)

\(=\)

\(\ds b\)

Sine of Right Angle, Cosine of Right Angle

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds b\)

Hence the result.

$\blacksquare$

Also see

• Equation of Vertical Line