Exercises: Relative position of two lines

Determine the relative position of the lines $r: y = 3x - 5$ and $s: y = -2x + 4$, justifying your answer based on their slopes.

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

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

Find the explicit equation of the line parallel to $r: y = -4x + 1$ that passes through the point $P(0, 3)$.

Check if the lines given in continuous form $r: \frac{x-1}{2} = \frac{y+2}{3}$ and $s: \frac{x}{4} = \frac{y-1}{6}$ are parallel or intersecting.

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

Given the lines $r: x = 2 - t, y = 3 + 2t$ and $s: 2x + y - 7 = 0$, study their relative position by comparing their direction vectors and checking if a point on $r$ belongs to $s$.

Find the point of intersection of the line $r: 3x - 2y - 12 = 0$ with the x-axis.

Two main irrigation pipes in a crop field follow the paths of the lines $r: 5x + 3y - 15 = 0$ and $s: 2x - y + 5 = 0$. Find out if they cross and, if so, find the exact coordinates of the connection point.

A straight road follows the equation $2x - y + 4 = 0$. A telephone repeater is located at the point $P(5, 1)$. Find the general equation of the shortest wiring line (which must be perpendicular) connecting the repeater to the road.

On the architectural plan of a building, two load-bearing walls are given by the equations $r: ax - 6y + 12 = 0$ and $s: 2x + by - 4 = 0$. Determine the values of $a$ and $b$ so that both walls represent exactly the same line (coincident lines).

A rescue drone moves along the straight line passing through the points $A(-2, 1)$ and $B(4, 5)$. Another surveillance drone follows the path given by the line $s: y = \frac{2}{3}x + 4$. Determine whether their paths are parallel or if they intersect at some point.

Two ships sail with paths given by the parametric equations $r: x = 1 + 2t, y = -3 + t$ and $s: x = -1 + 4k, y = 2 + 2k$. Check analytically if their routes are parallel or if they sail along the same line.

Calculate the value of the parameter $m$ so that the lines $r: mx + 2y - 5 = 0$ and $s: 3x - y + 1 = 0$ are perpendicular.

In a 3D design program, a laser cuts a metal plate along the line $r: 4x - 3y - 12 = 0$. Find the general equation of another cutting line, parallel to the previous one, that passes through the point $Q(-1, 2)$.

Two comets follow rectilinear paths in a 2D space model. The first passes through the points $P(0, 4)$ and $Q(3, 0)$. The second has the continuous equation $\frac{x-1}{-3} = \frac{y-2}{4}$. Check if their paths cross at any point or if they are parallel.

Three streets form a triangular park in the center of a city. The equations of the streets are $r: x + y - 6 = 0$, $s: 2x - y - 3 = 0$ and $t: x - 2y + 6 = 0$. Calculate the exact coordinates of the three vertices of the park.

Determine the relative position of the lines $r: kx + y - 3 = 0$ and $s: 4x + ky - 6 = 0$ depending on the possible values of the parameter $k$.

A robot vacuum cleaner moves along the path described by the line $r: 3x + 4y - 24 = 0$. Upon encountering an obstacle exactly at the point of intersection with the y-axis, it turns $90^{\circ}$ and continues its journey in a straight line. Find the general equation of the new path and the point where it will intersect the x-axis.

Find the coordinates of the symmetric point of $A(2, 5)$ with respect to the line $r: x - 2y + 3 = 0$.

The sides of a quadrilateral-shaped property are located on the lines $r: 2x + y - 5 = 0$ and $s: x - 2y + 5 = 0$. Knowing that one of the vertices of the property is the point $P(4, 2)$ and that the enclosure is a perfect rectangle, find the general equations of the lines that contain the other two sides.

Calculate the area of the triangle formed by the line $r: 5x + 2y - 20 = 0$ and the two coordinate axes.

Given the lines $r: 2x - y + 4 = 0$ and $s: x + y - 1 = 0$, find the explicit equation of a line parallel to $r$ that passes through the point of intersection of $s$ with the x-axis.

A ray of light follows the path of the line $r: x - y + 2 = 0$ and strikes a flat mirror located along the x-axis. Knowing that the angle of incidence is equal to the angle of reflection with respect to the normal, find the general equation of the line describing the reflected ray.