Relative position of two lines
Two roads in Dublin may never cross (parallel), cross at a single point (intersecting) or actually be the same road seen under another name (coincident). Two lines in the plane can only be in one of those three situations.
Three possible positions
Write the relative position of the lines $r: y = 3x - 5$ and $s: y = -2x + 4$, justifying your answer using their gradients.
Write the relative position of the lines $r: y = 3x - 5$ and $s: y = -2x + 4$, justifying your answer using their gradients.
The criterion using the coefficients
Determine the relative position of the lines $r: 2x - 4y + 6 = 0$ and $s: -x + 2y - 3 = 0$ by comparing their coefficients.
Determine the relative position of the lines $r: 2x - 4y + 6 = 0$ and $s: -x + 2y - 3 = 0$ by comparing their coefficients.
Calculate the value of the parameter $k$ so that the lines $r: 3x + ky - 2 = 0$ and $s: x - 2y + 5 = 0$ are parallel.
Calculate the value of the parameter $k$ so that the lines $r: 3x + ky - 2 = 0$ and $s: x - 2y + 5 = 0$ are parallel.
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 coordinates of the point of intersection of the intersecting lines $r: x + y - 4 = 0$ and $s: 2x - y - 5 = 0$.
Calculate the coordinates of the point of intersection of the intersecting lines $r: x + y - 4 = 0$ and $s: 2x - y - 5 = 0$.
Find the point of intersection of the line $r: 3x - 2y - 12 = 0$ with the x-axis (axis of abscissas).
Find the point of intersection of the line $r: 3x - 2y - 12 = 0$ with the x-axis (axis of abscissas).
Parallel or intersecting lines given in other forms
The same idea works for lines in any form: find the slope of each line (for $y = mx + c$ it is $m$; from two points it is rise ÷ run) and compare. Write each line as $y = mx + c$, then compare slopes and y-intercepts: equal slopes with different y-intercepts mean parallel, equal slopes with equal y-intercepts mean coincident.
Parametric form
A line can also be given by $x = x_0 + at,\ y = y_0 + bt$, where $t$ is a number that can take any value. The line passes through $(x_0, y_0)$ and has direction $(a, b)$, so its slope is $\frac{b}{a}$. To compare two lines, compare their directions: if $(a_1, b_1)$ and $(a_2, b_2)$ are multiples of each other, the lines are parallel or coincident. Then check whether a point of one line lies on the other.
If you want to test a point of one line on a line given by $ax+by+c=0$, substitute $x$ and $y$ in terms of $t$: if the equation holds for every $t$ the lines coincide, if it never holds they are parallel, and if it holds for one value of $t$ they intersect at that point.
Find the explicit equation of the line parallel to $r: y = -4x + 1$ passing through the point $P(0, 3)$.
Find the explicit equation of the line parallel to $r: y = -4x + 1$ passing through the point $P(0, 3)$.
Check whether the lines given in symmetric form $r: \frac{x-1}{2} = \frac{y+2}{3}$ and $s: \frac{x}{4} = \frac{y-1}{6}$ are parallel or intersecting.
Check whether the lines given in symmetric form $r: \frac{x-1}{2} = \frac{y+2}{3}$ and $s: \frac{x}{4} = \frac{y-1}{6}$ are parallel or intersecting.
Given the lines $r: x = 2 - t, y = 3 + 2t$ and $s: 2x + y - 7 = 0$, study their relative position by comparing their direction vectors and checking whether a point on…
Given the lines $r: x = 2 - t, y = 3 + 2t$ and $s: 2x + y - 7 = 0$, study their relative position by comparing their direction vectors and checking whether a point on $r$ belongs to $s$.
Two ships sail along paths given by the parametric equations $r: x = 1 + 2t, y = -3 + t$ and $s: x = -1 + 4k, y = 2 + 2k$. Check analytically whether their routes are…
Two ships sail along paths given by the parametric equations $r: x = 1 + 2t, y = -3 + t$ and $s: x = -1 + 4k, y = 2 + 2k$. Check analytically whether their routes are parallel or if they sail along the same line.
Applications: junctions, paths and shapes
Two lines are perpendicular when $m_1 m_2 = -1$. This lets us find a line through a point at right angles to another line, and also the reflection of a point in a line.
Two main irrigation pipes in a field follow the paths of the lines $r: 5x + 3y - 15 = 0$ and $s: 2x - y + 5 = 0$. Find out whether they cross and, if so, find the exact…
Two main irrigation pipes in a field follow the paths of the lines $r: 5x + 3y - 15 = 0$ and $s: 2x - y + 5 = 0$. Find out whether they cross and, if so, find the exact coordinates of the connection point.
Consider the lines $r: 2x - y - 6 = 0$ and $s: x + y - 6 = 0$. a) Without solving the algebraic system, analyze the sign of the slopes and obtain the intersection points…
Consider the lines $r: 2x - y - 6 = 0$ and $s: x + y - 6 = 0$.
- a) Without solving the algebraic system, analyze the sign of the slopes and obtain the intersection points with the coordinate axes for each line. From that qualitative geometric information, reasonably estimate in which quadrant of the Cartesian plane the intersection point between $r$ and $s$ will occur.
- b) Formally verify that they are intersecting lines using the relationship between their coefficients and calculate the exact coordinates of their intersection point by solving the linear system.
- c) Assess whether the coordinates obtained confirm your previous estimate and explain why anticipating the quadrant helps detect possible sign errors in the algebraic calculation.
Three streets form a triangular park in the centre of a city. The equations of the streets are $r: x + y - 6 = 0$, $s: 2x - y - 3 = 0$ and $t: x - 2y + 6 = 0$. Calculate…
Three streets form a triangular park in the centre of a city. The equations of the streets are $r: x + y - 6 = 0$, $s: 2x - y - 3 = 0$ and $t: x - 2y + 6 = 0$. Calculate the exact coordinates of the three vertices of the park.