Relative position of two lines
Two streets in a city may never cross (parallel), cross at a single point (intersecting) or actually be the same street 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 criterion works with lines in explicit, parametric or continuous form: just convert them to general form first, or compare their direction vectors directly.
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.
Applications: junctions, paths and shapes
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.
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.