Mathematics · Ch 9 — Straight Lines

Horizontal and Vertical Lines

9.3.1

Horizontal and Vertical Lines

When a line is horizontal, every point on it has the same y-coordinate. If a horizontal line L is at a distance $a$ from the x-axis, then the y-coordinate of every point on L is either $a$ or $-a$ . Which one it is depends on whether the line lies above the x-axis (positive distance) or below it (negative distance).

Similarly, a vertical line has the same x-coordinate for all its points. If a vertical line is at a distance $b$ from the y-axis, then the x-coordinate of every point on it is either $b$ or $-b$ , depending on whether the line is to the right or left of the y-axis.

Note

The distance is always taken as a positive number. The sign in the equation tells you which side of the axis the line lies on.

Equation of a Horizontal Line

Consider a horizontal line L at a distance $a$ from the x-axis. The ordinate (y-coordinate) of every point on L is constant. If the line is above the x-axis, the ordinate is $a$ ; if below, it is $-a$ .

Therefore, the equation of the horizontal line is:

y = a \quad \text{or} \quad y = -a

The sign is chosen based on the position of the line relative to the x-axis.

y = \pm a

Equation of a Vertical Line

Now consider a vertical line at a distance $b$ from the y-axis. The abscissa (x-coordinate) of every point on this line is constant. If the line is to the right of the y-axis, the abscissa is $b$ ; if to the left, it is $-b$ .

Therefore, the equation of the vertical line is:

x = b \quad \text{or} \quad x = -b

The sign is chosen based on the position of the line relative to the y-axis.

x = \pm b

Example: Lines Through a Given Point

Example 4: Find the equations of the lines parallel to the axes and passing through the point $(-2, 3)$ .

Solution:

We need two lines: one horizontal (parallel to the x-axis) and one vertical (parallel to the y-axis), both passing through $(-2, 3)$ .

Horizontal line through $(-2, 3)$ :

Every point on a horizontal line has the same y-coordinate. Since the line passes through $(-2, 3)$ , the y-coordinate of every point on it is $3$ . Therefore, the equation is:

$y = 3$ …

Figure 9.8Horizontal lines y=±a and vertical lines x=±b

Fig. 9.8 — Horizontal lines y=±a and vertical lines x=±b

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Fig 9.8 is a simple coordinate-plane sketch split into two side-by-side panels, (a) and (b). Its purpose is to show the geometric meaning of the equations of horizontal and vertical lines — the most basic lines after the axes themselves.

Panel (a) shows the $x$ -axis and $y$ -axis. Two horizontal lines are drawn: one above the $x$ -axis, labelled $y = a$ , and one below it, labelled $y = -a$ . A dashed vertical segment runs from the origin $(0,0)$ straight up to the line $y = a$ , and another dashed segment runs straight down to $y = -a$ . Each dashed segment is marked with the length $a$ . The idea is immediate: every point on the upper line has $y$ -coordinate $a$ ; every point on the lower line has $y$ -coordinate $-a$ . The distance of each line from the $x$ -axis is exactly $|a|$ .

Panel (b) is the same idea rotated. Two vertical lines are drawn: one to the right of the $y$ -axis, labelled $x = b$ , and one to the left, labelled $x = -b$ . Dashed horizontal segments from the origin go right to $x = b$ and left to $x = -b$ , each marked with length $b$ . Every point on the right-hand line has $x$ -coordinate $b$ ; every point on the left-hand line has $x$ -coordinate $-b$ . The distance of each line from the $y$ -axis is $|b|$ .

The physical idea is that a line parallel to an axis is completely described by a single coordinate — the constant value of the other coordinate. A horizontal line never changes its $y$ ; a vertical line never changes its $x$ . The sign of the constant tells you which side of the origin the line lies on.

\text{Horizontal line at distance } |a| \text{ from the } x\text{-axis: } y = a \text{ or } y = -a

Figure 9.9Lines x=−2 and y=3 through (−2,3)

Fig. 9.9 — Lines x=−2 and y=3 through (−2,3)

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

The figure shows a standard Cartesian plane with four quadrants. Two lines are drawn: a vertical line labelled $x = -2$ and a horizontal line labelled $y = 3$ . Both lines are shown as double-arrowed indigo lines, meaning they extend infinitely in both directions. The point where they cross is marked as $(-2, 3)$ .

The physical idea is straightforward. A horizontal line is defined by the fact that every point on it has the same $y$ -coordinate. A vertical line is defined by the fact that every point on it has the same $x$ -coordinate. The figure makes this concrete: the horizontal line through $(-2, 3)$ is the set of all points whose $y$ -coordinate is 3, regardless of $x$ . The vertical line through the same point is the set of all points whose $x$ -coordinate is $-2$ , regardless of $y$ .

The textbook uses this figure to develop the equations of lines parallel to the axes. The key result is:

y = a \quad \text{or} \quad y = -a

In the specific case of the point $(-2, 3)$ :

The horizontal line through it is $y = 3$ (since the point is above the $x$ -axis, the sign is positive).
The vertical line through it is $x = -2$ (since the point is to the left of the $y$ -axis, the sign is negative). …

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