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 $s: \frac{x-2}{2} = \frac{y+1}{3}$.

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

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.

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

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.

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.

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.

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.

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.

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.

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 $m$.

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 $b$.

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?

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.

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 $t: x + 4y - 12 = 0$.

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.