Exercises: Relative position of two lines

Determine the relative position of the lines $r: 2x - 3y + 5 = 0$ and $s: -4x + 6y - 10 = 0$ by directly comparing their coefficients.

Given the lines $r: y = \frac{3}{2}x - 1$ and $s: y = \frac{3}{2}x + 4$, state their relative position, justifying your answer using the values of their gradients.

Study the relative position of the line $r$, which passes through the points $A(1, 2)$ and $B(3, 5)$, and the line $s: \frac{x-2}{2} = \frac{y+1}{3}$.

Calculate the exact coordinates of the point of intersection of the intersecting lines $r: x + 2y - 4 = 0$ and $s: 3x - y - 5 = 0$.

Find the value of the parameter $k$ so that the lines $r: kx + 4y - 2 = 0$ and $s: 3x - 2y + 5 = 0$ are strictly parallel.

Two exploration drones follow straight-line paths described by the equations $r: x - y + 2 = 0$ and $s: 2x + y - 8 = 0$. Calculate the coordinates of the point in the plane where their paths cross.

Determine the relative position of the lines given in parametric form $r: \begin{cases} x = 1 + 2t \\ y = -3 + t \end{cases}$ and $s: \begin{cases} x = 5 - 4\lambda \\ y = 2 - 2\lambda \end{cases}$.

Find the value of $m$ so that the lines $r: mx - 6y + 1 = 0$ and $s: 2x + 3y - 4 = 0$ intersect and are not parallel.

Lucía needs to determine the relative position of the lines $r: 4x - 6y + 3 = 0$ and $s: \dfrac{x - 1}{3} = \dfrac{y + 2}{2}$. On her exam, she writes the following reasoning:

“Line $r$ has slope $m_r = -\dfrac{4}{6} = -\dfrac{2}{3}$. For line $s$, its direction vector is $\vec{v}_s = (3, 2)$, so its slope is $m_s = \dfrac{2}{3}$. Since the slopes have different signs ($-\dfrac{2}{3} \ne \dfrac{2}{3}$), I conclude that the lines are intersecting and cross at a single point; if they had the same sign, they would be parallel”.

Identify and explain with reasoning the two errors (one algebraic calculation error and one conceptual error) that Lucía made. Then, solve the exercise correctly by determining the true relative position of both lines by comparing their direction vectors or their slopes.

Given the line $r: 3x - 4y + 12 = 0$, state and obtain the general equations of two different lines, $s$ and $t$, that inversely satisfy the following geometric conditions:

1. a) Line $s$ must be parallel to $r$ and pass exactly through the origin $(0, 0)$.
2. b) Line $t$ must be parallel to $r$, but its $y$-intercept must be three times the $y$-intercept of $r$.

Justify algebraically why none of the three lines ($r$, $s$, and $t$) is coincident with the others by analyzing the proportionality of their coefficients.

In a security system, two laser beams are modelled in a coordinate plane. The first beam follows the line $r: x + y - 6 = 0$ and the second passes through the emitters located at $P(2, 1)$ and $Q(-2, -1)$. Determine whether the laser beams are parallel, coincident, or intersecting. If they intersect, find the exact point of intersection.

The trajectories of two aeroplanes on a two-dimensional radar screen are given by $r: \begin{cases} x = 2 + 3t \\ y = 1 - t \end{cases}$ and $s: \begin{cases} x = -1 + 2\lambda \\ y = 4 + \lambda \end{cases}$. Check whether the routes cross and, if so, calculate the point of intersection of the trajectories in the plane.

Discuss the relative position of the lines $r: (m-1)x + 2y - 3 = 0$ and $s: 4x + (m+1)y - 6 = 0$ according to the possible values of the parameter $m$.

Calculate the value of $k$ so that the lines $r: 2x - y + 1 = 0$, $s: x + y - 4 = 0$ and $t: 3x + ky - 2 = 0$ intersect at exactly the same point in the plane.

An architect designs two straight walkways in a park. Walkway A follows the direction of vector $\vec{u} = (4, -2)$ and passes through the fountain located at $F(0, 5)$. Walkway B is given by the equation $x + 2y - 10 = 0$. Prove analytically that both walkways overlap, being in fact the same line.

Find the coordinates of the vertices of the triangle formed by the intersection of the three lines $r: x - y - 2 = 0$, $s: 2x + 3y - 9 = 0$ and $t: x + 4y - 12 = 0$.

Given the lines $r: \frac{x-1}{2} = \frac{y+2}{3}$ and $s: 3x - 2y + k = 0$, find out whether there is any value of the parameter $k$ for which the lines are coincident. Reason your answer analytically.

A ray of light in an optics experiment follows the path described by the line $r: x - 2y + 4 = 0$. A flat mirror is placed exactly on the x-axis ($y = 0$). Determine the coordinates of the point of incidence of the ray on the mirror.

Given the lines $r: 3x - 2y + 7 = 0$ and $s: \begin{cases} x = 1 - 4\lambda \\ y = 2 - 6\lambda \end{cases}$ (with $\lambda \in \mathbb{R}$), the goal is to determine their relative position and find their intersection point if it exists. To solve this, two strategies are proposed:

• Method A: Transform line $s$ into its general implicit equation $Ax + By + C = 0$ and check the proportionality of the coefficients $\frac{A_1}{A_2}$, $\frac{B_1}{B_2}$, and $\frac{C_1}{C_2}$.

• Method B: Directly substitute the parametric expressions for $x(\lambda)$ and $y(\lambda)$ of line $s$ into the general equation of $r$ and solve the resulting linear equation in $\lambda$.

Apply both methods step by step to classify the relative position of the lines, and justify which of the two procedures is more direct, efficient, and carries a lower risk of error in this particular case.

The Maritime Rescue center in Tarifa monitors two vessels in waters of the Strait using Cartesian coordinates (where $1\text{ unit} = 1\text{ nautical mile}$). A rescue boat navigates in a straight line following the path of line $r$, which passes through the point $A(-1, 3)$ and has the direction of the vector $\vec{u} = (2, -1)$. At the same time, a cargo ship maintains a route described by the equation $s: x + 2y - 15 = 0$.

1. a) Reasonably determine the relative position of both paths. Is there a collision risk between their paths if both vessels were to continue their course indefinitely? If they intersect, find the coordinates of the meeting point.
2. b) Due to a steering failure in the rudder, the cargo ship changes its path and begins to follow the line $s_k: kx + 6y - 10 = 0$. Calculate the exact value of $k$ so that the new course is strictly parallel to the rescue boat, and analyze whether there is any value of $k$ that makes both paths coincident.

Discuss the relative position of the lines $r: ax + y - b = 0$ and $s: 2x - ay + 4 = 0$ by conducting a thorough analysis depending on the simultaneous parameters $a$ and $b$.

Three straight roads are modelled on a topographic map using the equations $r: x + 2y - 4 = 0$, $s: 2x - y - 3 = 0$ and $t: ax + y - 2 = 0$. Determine the values of the parameter $a$ for which the three roads intersect at a single point or are parallel to each other, preventing a triangle bounded by them from forming.

A parallelogram has two of its sides on the lines $r: x - 2y + 4 = 0$ and $s: x - 2y - 6 = 0$. Another of its sides lies on the line $t: 2x + y - 2 = 0$. Knowing that one of the vertices of the parallelogram is the point $P(4, 4)$, calculate the equation of the line containing the fourth side and determine the exact coordinates of the four vertices.

Determine the value or values of $k$ so that the point of intersection of the intersecting lines $r: 2x + ky - 6 = 0$ and $s: x - y + 2 = 0$ lies on the angle bisector of the first and third quadrants ($y = x$).

Consider the family or pencil of lines defined by the mathematical expression $(2x - y + 1) + k(x + 2y - 3) = 0$. Prove that all the lines that make up this pencil intersect at the same fixed point and calculate its coordinates. Next, find the general equation of the line in this pencil that is parallel to $s: 3x - 4y + 5 = 0$.

A billiard ball follows the path given by the line $r: x - y + 2 = 0$ and bounces off the cushion represented by the line $s: x + 2y - 8 = 0$. Knowing that the angle of incidence is equal to the angle of reflection, calculate the coordinates of the exact point where the ball hits the cushion and find the equation of the line describing the path after the bounce.

Given the lines $r: \begin{cases} x = 1 + \lambda \\ y = 2 - 2\lambda \end{cases}$ and $s: ax + by - 5 = 0$, calculate the values of the parameters $a$ and $b$ so that both lines are identical. Subsequently, find the general equation of a line perpendicular to both that passes through the origin of coordinates.

The projected orbits of two comets in a spatial plane follow the paths $r: (m+1)x + my - 2 = 0$ and $s: mx + (m-1)y - 1 = 0$. Determine whether there is any moment (value of $m$) at which the paths are strictly parallel. For which values of $m$ do the orbits cross at a single point?

Consider the lines $r: x - 2y + 4 = 0$ and $s: 3x + y - 9 = 0$.

1. a) Estimate: Without solving the system of equations algebraically or plotting them on a millimeter scale, deduce with reasoning which quadrant of the Cartesian plane these two lines will necessarily intersect in, and prove why the abscissa of said intersection point must strictly belong to the open interval $(0, 3)$. Base your estimation exclusively on the signs of the slopes and on the intercepts of each line with the Cartesian axes.
2. b) Check: Analytically solve the system formed by the equations of $r$ and $s$ to determine the exact coordinates of the intersection point $P(x, y)$.
3. c) Assess: Compare the analytical result obtained with your initial estimation and reflect on how prior qualitative analysis allows you to detect inconsistencies in algebraic calculations.

In the affine plane, consider two fixed lines, $r: x - y - 1 = 0$ and $s: 2x + y - 5 = 0$, along with a third line that depends on a real parameter: $t_k: kx + y - 7 = 0$.

The goal is to investigate under what conditions the three lines enclose a triangle in the plane.

1. a) Explain deductively what the only two geometric causes are for which three coplanar lines (knowing that $r$ and $s$ are already intersecting) DO NOT form a triangle.
2. b) Set up the algebraic conditions for relative position to calculate the values of $k$ that make $t_k$ parallel to line $r$ or parallel to line $s$. Check also whether $t_k$ becomes coincident with either of them for any of those values.
3. c) Find the value of $k$ for which the three lines are concurrent, that is, they pass simultaneously through the same point.
4. d) Using the previous sections, precisely determine the set of values of $k \in \mathbb{R}$ for which the three lines DO determine a triangle.