When representing a straight line in the Cartesian plane, we can, in some cases, notice that it can be parallel to the Ox axis (perpendicular to the Oy axis) or parallel to the Oy axis (perpendicular to the Ox axis).
To differentiate the vertical from the horizontal, we will take the abscissa axis (Ox axis) as a reference. Therefore, the line that is perpendicular to the Ox axis will be considered the vertical line, so the one perpendicular to the Oy axis will be horizontal.
These two types of lines have elements that facilitate the identification of their equations, see:
• Horizontal lines
This type of straight line will not intersect the Ox axis, so one of the information we can conclude is that the calculation of its slope will always be equal to: m = tg180° = 0, and will intersect the Oy axis at any point (k) of equal coordinates to (0.k).
With the value of its slope plus a point belonging to this horizontal line, we can conclude that the equation of this line will always be equal to:
y-y0 = m (x - x0)
y - k = 0 (x - 0)
y - k = 0 - 0
y = k
• Vertical Lines
This type of straight line will not intersect the Oy axis, so one of the information we can conclude is that on the vertical line it will not be possible to calculate its slope, as tg90° does not exist. And it will intercept the Ox axis at any point (k) with coordinates equal to (k, 0).
Without the value of the slope it is not possible to determine the equation of the straight line by defining the fundamental equation, but since the vertical line will intersect the abscissa axis always and only at point k, we conclude that its equation will be equal The: x = k.
by Danielle de Miranda
Graduated in Mathematics
Brazil School Team
Analytical Geometry - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/retas-horizontais-verticais.htm