Parallel, Intersecting, and Coincident Lines
A city planner is drawing two straight bike paths on a map. Will they ever cross? Do they run side by side forever? Or are they secretly the same path drawn twice? In this section you will learn to answer that from the equations alone, without graphing anything.
Three ways two lines can meet
When both lines are in slope-intercept form $y = mx + b$, you can read the position straight off the equations.
Try it: drag the slider so the second line has the same slope as the first. What do you see?
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.
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.
Comparing coefficients in $Ax + By + C = 0$
Lines are often given in standard form $Ax + By + C = 0$. You could solve for $y$ every time, but there is a faster test. The slope of $Ax + By + C = 0$ is $m = -\dfrac{A}{B}$, so two lines have equal slopes exactly when $\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2}$.
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.
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.
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.
Direction vectors, parametric and symmetric forms
If a line comes in parametric form, as a pair of points, or in symmetric form, you do not need to convert it. Compare the direction vectors $\vec u_r = \langle a_1, b_1 \rangle$ and $\vec u_s = \langle a_2, b_2 \rangle$.
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Parametric form: $x = x_0 + at,\ y = y_0 + bt$ has direction vector $\langle a, b \rangle$ and passes through $(x_0, y_0)$.
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Symmetric form: $\dfrac{x - x_0}{a} = \dfrac{y - y_0}{b}$ has the same direction vector and point.
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Two points: the line through $A$ and $B$ has direction vector $\overrightarrow{AB} = B - A$.
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 $s: \begin{cases} x = 5 - 4\lambda \\ y = 2 - 2\lambda \end{cases}$.
Determine the relative position of the line $r$, which passes through the points $A(1, 2)$ and $B(3, 5)$, and the line $s: \frac{x-2}{2} = \frac{y+1}{3}$.
Determine the relative position of the line $r$, which passes through the points $A(1, 2)$ and $B(3, 5)$, and the line $s: \frac{x-2}{2} = \frac{y+1}{3}$.
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.…
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.
Finding the point of intersection
If two lines intersect, their common point satisfies both equations. So you find it by solving the system, exactly like in Algebra 1: substitution or elimination.
Solving the system also tells you the position, even if you did not know it in advance:
| What happens when you solve | Position |
|---|---|
| You get one value for each variable | intersecting |
| You get a false statement such as $0 = 4$ | parallel (no solution) |
| You get a true statement such as $0 = 0$ | coincident (infinitely many solutions) |
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 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…
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.
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…
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.
Mixing forms
Problems rarely give both lines in the same form. A reliable plan:
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Turn each line into the form that is easiest to compare: general form for the coefficient test, or direction vector + point for a quick check.
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Decide the position.
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If the lines intersect, solve the system.
To write a line through a point with a given slope or a given direction $\langle a, b \rangle$, use point-slope form $y - y_1 = m(x - x_1)$ with $m = \dfrac{b}{a}$, then move everything to one side.
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…
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…
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.
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…
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.
Position depending on a parameter
When an equation contains a parameter such as $m$, the position can change with $m$. Here is the standard procedure.
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Write the "same direction" condition $A_1B_2 - A_2B_1 = 0$ and solve it for the parameter. These are the critical values.
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For each critical value, substitute it into both equations and compare all three ratios: parallel or coincident?
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For every other value of the parameter, the lines intersect.
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 $m$.
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 $m$.
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…
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 $b$.
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…
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?
Three lines at once
Three lines in the plane can create a triangle, or fail to. Three lines are concurrent if they all pass through one point.
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To get the vertices of the triangle, intersect the lines two at a time (three pairs, three vertices).
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To test or force concurrency, find the intersection of two of the lines and plug that point into the third equation.
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No triangle forms when the lines are concurrent, or when at least two of them are parallel (or coincident).
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 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…
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.
Families of lines
Take two intersecting lines $L_1 = 0$ and $L_2 = 0$ and form $L_1 + kL_2 = 0$, where $k$ is any real number. Each value of $k$ gives a different line. (Some problems call this collection a pencil of lines; it is a family of lines.)
The same idea of "pinning the intersection point" helps when the problem says the intersection lies on a given line.
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 $s: 3x - 4y + 5 = 0$.
Geometric applications
Here are three tools that show up in harder problems.
Bouncing off a mirror (reflection)
When a ray hits a straight mirror, the angle of incidence equals the angle of reflection. A clean way to find the bounce: reflect a point of the incoming ray across the mirror. The reflected ray lies on the line through the hit point and that reflected point.
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…
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.
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…
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…
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.
Back to the start: you can now tell, from the equations alone, whether the two bike paths are parallel, cross at one point, or are the same path. You compare slopes, the coefficients $A$, $B$, $C$, or the direction vectors, and if they cross you solve the system to find exactly where.