Relative position of two lines
Picture two straight Luas tracks on a map of Dublin, two laser beams or two drone paths. They can only be related in three ways: they never cross, they cross at one point, or they are in fact the same line. Being able to tell which of the three it is, without drawing anything, is the first step in the coordinate geometry of the line for Leaving Cert Higher Level. At the end we will use it to decide whether two cycle paths in the Phoenix Park ever meet.
Three possible positions
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.
The criterion using the 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.
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.
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.
Aoife needs to determine the relative position of the lines $r: 4x - 6y + 3 = 0$ and $s: \dfrac{x - 1}{3} = \dfrac{y + 2}{2}$. On her exam, she writes the following…
Aoife needs to determine the relative position of the lines $r: 4x - 6y + 3 = 0$ and $s: \dfrac{x - 1}{3} = \dfrac{y + 2}{2}$. On her exam, she writes the following reasoning:
“Line $r$ has slope $m_r = -\dfrac{4}{6} = -\dfrac{2}{3}$. For line $s$, its direction vector is $\vec{v}_s = (3, 2)$, so its slope is $m_s = \dfrac{2}{3}$. Since the slopes have different signs ($-\dfrac{2}{3} \ne \dfrac{2}{3}$), I conclude that the lines are intersecting and cross at a single point; if they had the same sign, they would be parallel”.
Identify and explain with reasoning the two errors (one algebraic calculation error and one conceptual error) that Aoife made. Then, solve the exercise correctly by determining the true relative position of both lines by comparing their direction vectors or their slopes.
Lines given by a direction, a parametric form or two points
Sometimes a line is given by a point and a direction vector, in parametric form, in continuous form, or by two points on it. A direction vector $\vec{u}=(a,b)$ means: go $a$ across and $b$ up, so the slope is $m=\dfrac{b}{a}$. For two points, $m=\dfrac{y_2-y_1}{x_2-x_1}$. Then you can compare slopes as before, or compare the direction vectors $\vec{u}_r=(a_1,b_1)$ and $\vec{u}_s=(a_2,b_2)$ directly.
The graph shows the two lines of the example below: same slope $\tfrac12$, different y-intercepts.
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.
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 $F(0, 5)$.…
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.
Finding the point of intersection
If the lines are intersecting, their point of intersection is found by solving the system formed by the two equations.
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.
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.
Position as a function of a parameter
When one of the lines depends on a parameter, you impose the condition for parallelism (or for intersection) on the coefficients and solve for the value that satisfies it.
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 $s: \frac{x-2}{2} = \frac{y+1}{3}$.
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 $m$.
Parameters in both lines
When both lines depend on a parameter, the lines are parallel or coincident exactly when $A_1B_2-A_2B_1=0$ (the cross-multiplied form, which avoids dividing by something that might be zero). Solve this for the parameter (often a quadratic), then test each value with the $C$ coefficients to decide between parallel and coincident. For every other value the lines intersect.
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 $b$.
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?
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.…
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.
Three lines
Three lines can pass through one point (concurrent), form a triangle, or fail to form one. To find a missing parameter for concurrent lines, find the intersection of the two lines that have no parameter, then substitute that point into the third equation. Three lines do not form a triangle if all three go through one point, or if two of them are parallel. If a triangle does form, its vertices are the three pairwise intersections.
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.
Parallel, perpendicular and geometry problems
A line parallel to $Ax+By+C=0$ has the same $A$ and $B$: $Ax+By+C'=0$, and you find $C'$ by substituting the given point. Lines are perpendicular when $m_1m_2=-1$; a line perpendicular to $Ax+By+C=0$ has the form $Bx-Ay+C'=0$. Two lines are identical when all their coefficients are proportional, so scale the equation to match the given constant.
A pencil of lines $L_1+kL_2=0$ (where $L_1=0$ and $L_2=0$ are lines) contains lines that all go through the point where $L_1=0$ and $L_2=0$ both hold, whatever $k$ is. Solve $L_1=0$, $L_2=0$ to find it.
Given the line $r: 3x - 4y + 12 = 0$, state and obtain the general equations of two different lines, $s$ and $t$, that inversely satisfy the following geometric…
Given the line $r: 3x - 4y + 12 = 0$, state and obtain the general equations of two different lines, $s$ and $t$, that inversely satisfy the following geometric conditions:
- a) Line $s$ must be parallel to $r$ and pass exactly through the origin $(0, 0)$.
- b) Line $t$ must be parallel to $r$, but its $y$-intercept must be three times the $y$-intercept of $r$.
Justify algebraically why none of the three lines ($r$, $s$, and $t$) is coincident with the others by analyzing the proportionality of their coefficients.
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.
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 $s: 3x - 4y + 5 = 0$.
Back to the start: two cycle paths in the Phoenix Park are $y=\tfrac12x+1$ and $x-2y+8=0$. The second has slope $\tfrac12$ and y-intercept $4$. Same slope and different y-intercepts: the paths are parallel and never meet.