Relative position of two lines
Two straight-line paths —two streets, two laser beams, two drones— 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 all of the analytic geometry of Bachillerato.
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.
Comparing direction vectors
If the lines are given in parametric or continuous form, there is no need to convert them to general form: it is enough to compare their direction vectors $\vec{u}_r=(a_1,b_1)$ and $\vec{u}_s=(a_2,b_2)$.
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 $s: \begin{cases} x = 5 - 4\lambda \\ y = 2 - 2\lambda \end{cases}$.
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.
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 $m$.