Exercises: Lines parallel to the axes

Write down the equation of the line parallel to the x-axis ($X$-axis) that passes through the point $A(3, -5)$.

Write down the equation of the line parallel to the y-axis ($Y$-axis) that passes through the point $B(-2, 7)$.

State the direction vector and the gradient of the line with equation $y = 4$.

State the direction vector and the gradient of the line with equation $x = -1$.

Determine the coordinates of the point of intersection of the lines $r: x = 5$ and $s: y = -2$.

Write down the general equations of the coordinate axes $X$ and $Y$.

A line passes through the points $P(4, 2)$ and $Q(-3, 2)$. Is it parallel to either of the axes? Write down its equation.

Given the line in parametric form $x = -3$, $y = t$, state which axis it is parallel to and write down its general equation.

A delivery drone moves through a city with grid-like streets. Its path follows a straight line parallel to the $Y$-axis passing through the coordinate $(6, 2)$. Write down the equation of its path and the coordinates of two points it will pass through.

The printhead of a 3D printer must draw a constant horizontal support line at $15\text{ mm}$ from the origin on the $Y$-axis. Write down the parametric equations and the general equation of this line.

Determine the value of $k$ so that the line with general equation $kx + 3y - 6 = 0$ is parallel to the $X$-axis. Write down the resulting equation.

Determine the value of $a$ so that the line with equation $4x + (a+1)y - 8 = 0$ is parallel to the $Y$-axis. Write down the resulting equation.

In a chemistry experiment, the temperature $T$ (in degrees Fahrenheit) is kept constant at $35^\circ\text{C}$ over time $t$ (in minutes). Representing time on the $X$-axis and temperature on the $Y$-axis, write down the equation of the line and its direction vector.

Find the equation of the line perpendicular to $x = 7$ that passes through the point $M(-4, 9)$.

On a soccer field, we represent the center of the field at the origin $(0,0)$. If the field is $100\text{ m}$ long (located on the $X$-axis) and $60\text{ m}$ wide (located on the $Y$-axis), write down the equations of the four straight lines that bound the playing field.

Calculate the minimum distance separating the parallel lines $r: y = 8$ and $s: y = -3$.

A drip irrigation system is installed along a pipe parallel to the $Y$-axis. We know that the pipe must water a tree located at the point $A(5, 12)$ and that the exact distance from the pipe to the $Y$-axis is $5\text{ m}$. Write down the equation of the line representing the pipe.

Given the triangle with vertices $A(-3, 2)$, $B(5, 2)$ and $C(1, 8)$, find the equation of the altitude from vertex $C$. Which axis is this altitude parallel to?

Find the value of $m$ so that the line $(m^2-9)x + (m+3)y + 5 = 0$ is parallel to the $X$-axis. Is there any value of $m$ for which this same line is parallel to the $Y$-axis? Justify your answer.

In the development of a video game, the screen has a resolution of $1920 \times 1080$ pixels. If we place the origin $(0,0)$ at the bottom-left corner, write the equations of the two lines (one horizontal and one vertical) that divide the screen exactly into four equal quadrants.

A square has its center at the point $C(3, 3)$ and its sides are parallel to the coordinate axes. If the side of the square measures $4\text{ u}$, write the equations of the four lines that contain its sides.

A robotic arm must move from the point $P(-2, -5)$ to the point $Q(4, 7)$ following a stepped path composed of two segments: first a movement parallel to the $X$-axis and then a movement parallel to the $Y$-axis. Write the equations of the lines that contain these two segments.

Determine the relative position of the line $r: x = 4$ and the line $s: 2x + ky - 8 = 0$ depending on the parameter $k$.

An electronic components factory has fixed monthly costs of $12000\text{ \$}$, regardless of the quantity $x$ of parts manufactured. The revenue is given by the line $y = 30x$. Write the equation of the fixed cost line on the $(x, y)$ plane and find the exact point where the revenue equals the fixed costs.