Relative Position of Two Lines
Two Metrobús lines, two laser beams, two drones flying over Mexico City: if their paths are straight lines, there are only three possibilities. They never cross, they cross at one point, or they turn out to be the same path. Can you tell which of the three it is without drawing anything? With equations, you can.
Three possible positions
The criterion using slopes and coefficients
If you solve for $y$ in each line you get $y=mx+b$. Two lines with the same inclination ($m$) are parallel or the same line; with different inclinations, sooner or later they cross.
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.
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.
Determine the relative position of the lines $r: 2x - 3y + 5 = 0$ and $s: -4x + 6y - 10 = 0$ by directly comparing their coefficients.
Determine the relative position of the lines $r: 2x - 3y + 5 = 0$ and $s: -4x + 6y - 10 = 0$ by directly comparing their coefficients.
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 the parameter $k$ so that the lines $r: kx + 4y - 2 = 0$ and $s: 3x - 2y + 5 = 0$ are strictly parallel.
Comparing direction vectors
If the lines are given in parametric or symmetric (continuous) form, there is no need to convert them to general form: you compare their direction vectors $\vec{u}_r=(a_1,b_1)$ and $\vec{u}_s=(a_2,b_2)$. If you are given two points, the direction vector is the difference of their coordinates.
Study the relative position of the line $r$, which passes through the points $A(1, 2)$ and $B(3, 5)$, and the line $s: \frac{x-2}{2} = \frac{y+1}{3}$.
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…
Determine the relative position of the lines given in parametric form $r: \begin{cases} x = 1 + 2t \\ y = -3 + t \end{cases}$ and $s: \begin{cases} x = 5 - 4\lambda \\ y = 2 - 2\lambda \end{cases}$.
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…
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.
Finding the intersection point
If the lines are intersecting, the intersection point satisfies both equations at once: you find it by solving the system, using any of the methods you already know (equating, substitution, elimination). If the equations are parametric, it is best to convert them to general form first.
Calculate the exact coordinates of the point of intersection of the intersecting lines $r: x + 2y - 4 = 0$ and $s: 3x - y - 5 = 0$.
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…
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.
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…
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.
Lines with a parameter
When a line depends on a parameter, you impose the condition the problem asks for on the coefficients and solve.
If there are two parameters (such as $a$ and $b$), the procedure is the same: first the equal-directions condition, $A_1B_2=A_2B_1$, which usually gives values for just one parameter; then, for each one, you check the $C$ (which may involve the other parameter).
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.
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 $r: (m-1)x + 2y - 3 = 0$ 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 $a$…
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 lines
Three lines can pass through the same point, form a triangle, or have a pair of parallel lines.
To find the vertices of the triangle formed by three lines, you solve the three systems of two equations (each pair gives one vertex). No triangle forms when all three pass through one point or when two are parallel.
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.
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 $t: x + 4y - 12 = 0$.
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…
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.
Parallels through a point and pencils of lines
Parallel through a point. A line parallel to $Ax+By+C=0$ has the same $A$ and $B$ and only $C$ changes: $Ax+By+C'=0$. To find $C'$, substitute the point the line must pass through.
For the parallelogram: opposite sides are parallel. The fourth side is parallel to $t$ and passes through the vertex that is not on $t$. Each vertex is the intersection of two sides.
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…
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…
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 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…
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.
Perpendicular lines and reflection
To find out whether there is an $m$ that gives strictly parallel paths (as in the comet exercise), impose $A_1B_2=A_2B_1$, solve for $m$, and check the $C$ in each solution.
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…
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.
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…
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.
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…
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?
Back to the beginning: two straight paths are classified by comparing slopes, coefficients, or direction vectors; if they are intersecting, the system gives the crossing point.