Lines parallel to the axes

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

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

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

Design three points on the Cartesian plane, called $A$, $B$, and $C$, that satisfy all three of the following conditions at the same time:

1. The line passing through $A$ and $B$ is parallel to the $X$-axis.

2. The line passing through $B$ and $C$ is parallel to the $Y$-axis.

3. The intersection point between both lines is $B(-3, 4)$ and the distance between point $A$ and point $B$ is $6$ units.

Write all possible coordinates for point $A$, propose valid coordinates for point $C$, and write the equation of each of the two lines you have determined.

Consider the fixed lines $r: y = 6$, $s: y = 1$, and $t: x = 2$. We are looking for a fourth vertical line $u: x = k$ (with $k \neq 2$) such that the four lines delimit a rectangular region with an area equal to $20\text{ square units}$.

1. a) Before calculating: observe the distance separating lines $r$ and $s$. Mentally estimate what length the base of the rectangle must have and predict how many different values $k$ can take, indicating which sides of line $t$ they will be located on.
2. b) State the mathematical relationship between the area, the fixed height, and the base in terms of the parameter $k$. Solve the equation to find the exact values of $k$ and write the equations of the possible lines $u$.
3. c) Check whether the values obtained match your previous estimation and geometrically explain the meaning of there being exactly two solutions.

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

Seán has to find the equation of the line passing through points $A(5, -2)$ and $B(5, 7)$ on an exam. On his answer sheet, he writes the following:

"I calculate the slope: $m = \frac{7 - (-2)}{5 - 5} = \frac{9}{0} = 0$. Since the slope is $0$, the line is horizontal and its equation is $y = 5$, because the first coordinate of both points is $5$".

1. a) Identify the two conceptual errors made by Seán (one concerning the mathematical operation performed and another concerning the relationship between the coordinates and the orientation of the line).
2. b) Explain how the correct equation is deduced by directly observing the coordinates of $A$ and $B$, and write that equation.

To find the equation of the line passing through the points $P(-7, 3)$ and $Q(5, 3)$, two 4th Year students propose different strategies:

• Aoife says: "I apply the two-point form of the line: $\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1}$, substitute the coordinates, and cross-multiply to solve."

• Cian says: "There is no need to calculate: I notice that the ordinate of both points is the same ($y = 3$), so the line does not go up or down, it is horizontal, and its equation is directly $y = 3$."

1. a) Justify why Cian's reasoning is correct and explain what algebraic drawback Aoife would encounter if she tried to apply her formula to a vertical line, such as the one joining $R(2, -1)$ and $S(2, 8)$.
2. b) In light of the above, argue which of the two approaches is safer and more efficient for identifying lines parallel to the axes.

The head of a 3D printer must trace a constant horizontal support line at $15\text{ mm}$ from the origin on the $Y$ axis. Write 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 the resulting equation.

Consider the family of lines given by the general implicit equation:

where $a$ and $b$ are constant real parameters.

1. a) Determine the exact numerical values of $a$ and $b$ so that the line is parallel to the abscissa axis ($X$-axis) and passes through the point $P(7, -5)$. Write the resulting equation.
2. b) Analyze whether it is possible to adjust the parameters $a$ and $b$ so that the resulting line is parallel to the ordinate axis ($Y$-axis) and, in addition, coincides exactly with the $Y$-axis itself. Justify your conclusion using the algebraic conditions that the coefficients must satisfy.

On a football pitch, we represent the centre of the pitch at the origin of coordinates $(0,0)$. If the pitch is $100\text{ m}$ long (along the $X$ axis) and $60\text{ m}$ wide (along the $Y$ axis), write the equations of the four straight lines that bound the pitch.

Geological Survey Ireland (GSI) establishes a rectangular security perimeter around a hazardous area on the coast of County Kerry. A monitoring station is taken as the origin of coordinates $(0, 0)$, measuring distances in kilometers ($X$-axis to the east, $Y$-axis to the north). The perimeter has its sides oriented exactly along the cardinal directions, and two of its opposite vertices are $V_1(-4, -2)$ and $V_2(3, 5)$.

1. a) Write the equation of each of the four lines that form the boundaries of the perimeter, indicating which cardinal direction (north, south, east, or west) each one corresponds to.
2. b) An emergency patrol sends its GPS position from the point $P(-4, 1)$. Reasonably determine whether the patrol is inside the exclusion zone, outside of it, or exactly on one of its boundaries.

A square has its centre at 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 containing its sides.