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 $s: y = \frac{3}{2}x + 4$, indicate their relative position, justifying your answer based on the value of their slopes.

Determine 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 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 $s: 2x + 3y - 4 = 0$ are intersecting and not parallel.

In a security system, two laser beams are modeled on 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 if they cross. If they cross, find the exact point of intersection.

The paths of two airplanes 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 paths in the plane.

Determine the relative position of the lines $r: (m-1)x + 2y - 3 = 0$ and $s: 4x + (m+1)y - 6 = 0$ depending on 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 the 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 reality 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 if there is any value of the parameter $k$ for which the lines are coincident. Justify your answer algebraically.

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.

Determine the relative position of the lines $r: ax + y - b = 0$ and $s: 2x - ay + 4 = 0$ by carrying out an exhaustive analysis depending on the simultaneous parameters $a$ and $b$.

Three straight roads are modeled on a topographic plane by 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 being formed.

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 is on the line $t: 2x + y - 2 = 0$. Given that one of the vertices of the parallelogram is the point $P(4, 4)$, find 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 trajectory 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 that describes the trajectory 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. Then, find the general equation of a line perpendicular to both that passes through the origin.

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