Exercises: Parallelism and perpendicularity of lines

Write down the gradient of a line that is parallel to and the gradient of a line that is perpendicular to the line with equation $y = 4x - 7$.

Given the lines $r: y = -2x + 5$ and $s: y = \frac{1}{2}x - 3$, state with reasons whether they are parallel, perpendicular, or intersecting but non-perpendicular.

Find the explicit equation of the line that is parallel to $y = -3x + 2$ and passes through the point $P(0, 5)$.

A direction vector of line $r$ is $\vec{v} = (3, -1)$. Write down a direction vector for a line parallel to $r$ and another for a line perpendicular to $r$.

Determine the point-slope equation of the line perpendicular to $y = \frac{2}{5}x - 1$ passing through the point $A(4, 2)$.

Check analytically whether the lines given in general form $r: 3x - 6y + 2 = 0$ and $s: x - 2y - 5 = 0$ are parallel.

Write down the general equation of the line parallel to the x-axis passing through point $Q(-3, 8)$.

Calculate the gradient of the line passing through points $A(2, -1)$ and $B(5, 8)$, and deduce from it what the gradient of any line perpendicular to it would be.

Seán has to find the slope-intercept form of line $s$, perpendicular to line $r: y = -\dfrac{3}{4}x + 2$, passing through the point $P(0, -5)$. On his exam, he writes the following:

"Since it is perpendicular, I change the sign of the slope, so $m_s = \dfrac{3}{4}$. Since it intersects the y-axis at $-5$, the line is $y = \dfrac{3}{4}x - 5$".

1. a) Identify and explain with reasoning Seán's conceptual error. Rely on the geometric meaning of the slope (run and rise) to justify why his line does not form a $90^\circ$ angle with $r$.
2. b) Correct the solution and write the correct slope-intercept form of line $s$.

Construct the general equations ($Ax + By + C = 0$) of two distinct lines, $r$ and $s$, that simultaneously satisfy the following conditions:

1) Both must be perpendicular to the given line $t: 2x - 5y + 10 = 0$.

2) Line $r$ must pass through the origin $(0, 0)$.

3) Line $s$ must intersect the x-axis at the point $(4, 0)$.

Once obtained, prove that $r$ and $s$ are parallel to each other by analyzing their slopes or their coefficients, and explain the geometric meaning of the difference between their constant terms $C$.

On a city map, O'Connell Street follows the path of the line $2x + 5y - 10 = 0$. Grafton Street is perpendicular to it and passes through a square located at coordinates $P(-2, 3)$. Find the general equation of Grafton Street.

Calculate the value of parameter $k$ so that the line $kx - 3y + 6 = 0$ is parallel to the line joining points $A(1, 2)$ and $B(4, 11)$.

A laser beam travels along the line $y = 3x - 2$. At the exact point where the laser crosses the y-axis, a mirror is placed perpendicular to the ray's path. Determine the explicit equation of the line representing the mirror.

The adjacent sides of a rectangular plot lie on the lines $r: 2x - 4y + 7 = 0$ and $s: 4x + 2y - 3 = 0$. Check analytically that the sides form a right angle. Then, find the equation of a line parallel to $r$ that passes through the origin.

Two aeroplanes fly at the same altitude. The route of the first aeroplane is given by the parametric equations $x = 1 + 5t$, $y = -2 - 3t$. The route of the second aeroplane is parallel to that of the first and passes through the control tower $T(0, 0)$. Write down the parametric equations of the second aeroplane's route.

Determine the value of parameter $m$ so that lines $r: mx + 4y - 1 = 0$ and $s: 2x - 5y + 8 = 0$ are perpendicular.

A main irrigation pipe follows the line given by the symmetric equation $\frac{x-2}{4} = \frac{y+1}{-3}$. A secondary pipe perpendicular to the main line needs to be installed to connect to a sprinkler located at $S(-1, 5)$. Find the symmetric equation of this secondary pipe.

Find the explicit equation of the perpendicular bisector of the segment with endpoints $A(-2, 3)$ and $B(4, 1)$. (Remember that the perpendicular bisector is the line perpendicular to the segment that passes through its midpoint).

Two telecommunications repeaters are located at coordinates $A(-2, -1)$ and $B(4, 3)$. To distribute coverage evenly, technicians must plot the perpendicular bisector of the segment $AB$.

1. a) Before performing any algebraic calculations:

• Reasonably deduce whether the slope of segment $AB$ is positive or negative and, from it, what sign the slope of the perpendicular bisector must have.

• Mentally determine which quadrant the midpoint of segment $AB$ will be in, and estimate whether the y-intercept of the perpendicular bisector will be positive or negative.

1. b) Carry out the analytical calculations: find the coordinates of the midpoint of $AB$, the slope of the perpendicular bisector, and write its equation in general form ($Ax + By + C = 0$).
2. c) Check whether the values calculated in part b confirm the estimates you made in part a, and explain why the result is geometrically consistent.

In the railway network, a straight section of high-speed track follows the path of the line $r: 3x + 4y - 24 = 0$ (where the coordinates $x$ and $y$ are in kilometres relative to a control centre). A logistics platform located at point $P(7, 7)$ needs to be connected to the main track by means of a completely straight secondary branch line.

1. a) To minimize cost and environmental impact, the branch line must follow the shortest possible distance to the track. Find the general equation of the line $s$ along which this branch line must run, and explain what geometric property ensures that it is the shortest path.
2. b) Calculate the exact coordinates of the connection point $Q$ where the branch line will join the main track by solving the system formed by both lines.
3. c) Regulations establish that if the distance between the platform and the main track exceeds $4\text{ km}$, an intermediate technical station must be built. Justify whether this station will be necessary by calculating the exact distance between $P$ and $Q$.

The roof design of a sports hall includes two steel beams that cross each other. Beam 1 lies along the line $ax + 3y - 5 = 0$ and beam 2 along the line $2x - by + 4 = 0$. Calculate the exact values of $a$ and $b$ knowing that the beams are perpendicular and that beam 1 passes through point $P(1, -2)$.

Prove analytically that the quadrilateral whose vertices are $A(1, 1)$, $B(5, 3)$, $C(4, 5)$ and $D(0, 3)$ is a rectangle. To do this, check the gradients of its sides and confirm that adjacent angles are right angles.

A ship sails in a straight line following the path $r: 3x + 4y - 24 = 0$. A lighthouse is located at point $F(7, 7)$. Calculate the shortest distance from the lighthouse to the ship's path. (Hint: find the line perpendicular to $r$ that passes through $F$, find the point of intersection and calculate the distance between both points).

Given the lines $r: \frac{x+1}{2} = \frac{y-3}{k}$ and $s: 3x - 5y + 2 = 0$, determine the value of $k$ so that they are parallel. Next, calculate the value of $k$ so that they are perpendicular.

Calculate the exact coordinates of the orthocentre (point of intersection of the three altitudes) of the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(2, -5)$.

A solar panel is aligned with the line passing through the points $P(1, 4)$ and $Q(5, 2)$. To maximise efficiency, the sun's rays must strike the panel perpendicularly. Write the parametric equations of the path of a sunbeam that strikes exactly at the midpoint of the panel.

Find the exact coordinates of the orthogonal projection of the point $P(3, -2)$ onto the line $r: x - 2y + 5 = 0$. (Note: the orthogonal projection is the point of intersection of the line $r$ with the line perpendicular to it passing through $P$).

Two robot vacuum cleaners move across an open-plan room. The first follows the path given by the equation $y = \frac{1}{2}x + 2$. The second moves along a line perpendicular to that of the first, such that both would collide at the exact point of intersection of the first path with the y-axis (axis of ordinates). Find the explicit equation of the second robot's path and calculate the area of the triangle formed by the paths of both robots and the x-axis (axis of abscissas).

To check whether the lines $r: 12x - 15y + 8 = 0$ and $s: 10x + 8y - 3 = 0$ are perpendicular to each other, two students propose different approaches:

• Method 1: Solve for the explicit equation ($y = mx + n$) of each line, identify their slopes, and check if $m_r \times m_s = -1$.

• Method 2: Directly apply the condition on the coefficients of the general equation ($A_r \times A_s + B_r \times B_s = 0$).

1. a) Solve the problem using both methods and verify that they lead to the same conclusion.
2. b) Compare the two approaches: which one is more efficient and minimizes the risk of making errors with fractions and signs? Algebraically deduce why the condition $A_r \times A_s + B_r \times B_s = 0$ is exactly equivalent to $m_r \times m_s = -1$.
3. c) Knowing that $r$ and $s$ are perpendicular, is it possible to find any line in the plane that is parallel to $r$ and not perpendicular to $s$? Justify your answer logically without needing to write numerical equations.

On an urban plan, three buildings $A$, $B$, and $C$ are to be located so that they form a right-angled triangle with the right angle at vertex $B$. The positions of the first two are fixed: $A(-1, 2)$ and $B(3, 5)$. The third building, $C$, must be located on an avenue whose path is the horizontal line with equation $y = 11$.

1. a) State the analytical condition of perpendicularity between the lines containing sides $AB$ and $BC$, and calculate the exact coordinates $(x_C, y_C)$ where building $C$ must be built.
2. b) Now suppose that the project is changed and building $C$ must necessarily be located on a vertical street with equation $x = k$. Algebraically prove whether there is any specific value of $k$ for which it is completely impossible to build the right-angled triangle with a right angle at vertex $B$, and explain what anomalous geometric situation occurs in that case.