Exercises: Relative position of two lines

Write the relative position of the lines $r: y = 3x - 5$ and $s: y = -2x + 4$, justifying your answer using their gradients.

Determine the relative position of the lines $r: 2x - 4y + 6 = 0$ and $s: -x + 2y - 3 = 0$ by comparing their coefficients.

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 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.

Calculate the value of the parameter $k$ so that the lines $r: 3x + ky - 2 = 0$ and $s: x - 2y + 5 = 0$ are parallel.

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 $s$.

Find the point of intersection of the line $r: 3x - 2y - 12 = 0$ with the x-axis (axis of abscissas).

Cian has to determine the relative position of the lines $r: 2x - 3y + 6 = 0$ and $s: 4x - 6y - 10 = 0$ in class. In his notebook, he writes:

"I calculate the ratio of the coefficients: $\frac{2}{4} = \frac{-3}{-6} = \frac{1}{2}$. Since it gives the same result, the two lines have exactly the same points and are therefore coincident."

1. a) Identify the conceptual error that Cian makes and explain what check he forgot to perform.
2. b) Correct his solution by determining the correct relative position using the coefficients criterion.
3. c) What value should the constant term of line $s$ have for Cian's conclusion to be truly correct? Justify your answer.

Given the line $r: 3x - 4y + 12 = 0$:

1. a) Write the general equation of a line $s$ that is parallel to $r$ and passes through the origin $(0, 0)$. Justify its parallelism by comparing the ratios of their coefficients.
2. b) Find the coordinates of the intersection point of line $r$ with the x-axis ($y = 0$).
3. c) Write the general equation of a line $t$ that intersects $r$ and whose intersection with it is precisely that intersection point with the x-axis. Explain how you chose the coefficients of $t$ to ensure that it is not parallel to $r$.

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.

A straight road follows the equation $2x - y + 4 = 0$. A mobile phone mast is located at point $P(5, 1)$. Find the general equation of the shortest cabling line (which must be perpendicular) connecting the mast to the road.

On the architectural floor plan of a building, two load-bearing walls are given by the equations $r: ax - 6y + 12 = 0$ and $s: 2x + by - 4 = 0$. Determine the values of $a$ and $b$ so that both walls represent exactly the same line (coincident lines).

A rescue boat moves along the straight line passing through the points $A(-2, 1)$ and $B(4, 5)$. A sailboat follows the path given by the line $s: y = \frac{2}{3}x + 4$. Determine whether their paths are parallel or if, on the contrary, they intersect at some point.

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.

Calculate the value of the parameter $m$ so that the lines $r: mx + 2y - 5 = 0$ and $s: 3x - y + 1 = 0$ are perpendicular.

In a 3D design program, a laser cuts a metal sheet along the line $r: 4x - 3y - 12 = 0$. Find the general equation of another cut line, parallel to the previous one, passing through the point $Q(-1, 2)$.

Two comets follow straight paths in a 2D space model. The first passes through the points $P(0, 4)$ and $Q(3, 0)$. The second has the symmetric equation $\frac{x-1}{-3} = \frac{y-2}{4}$. Check whether their paths intersect at any point or if they are parallel.

To find out the relative position of the lines $r: y = -\frac{3}{4}x + 2$ and $s: 6x + 8y - 5 = 0$, two students propose different approaches:

• Aoife proposes transforming line $r$ into its general form ($Ax + By + C = 0$) and directly applying the coefficient proportionality criterion ($\frac{A_1}{A_2}$, $\frac{B_1}{B_2}$, $\frac{C_1}{C_2}$).

• Seán prefers to solve for the variable $y$ in the equation of $s$ to leave it in slope-intercept form ($y = mx + n$) and compare slopes and y-intercepts.

1. a) Apply Aoife's method and conclude the relative position of both lines.
2. b) Apply Seán's method and verify that you reach the same conclusion.
3. c) Compare both procedures: explain which of the two you consider more direct or with less risk of making calculation errors (signs or fractions) for this specific pair of equations.

Consider the lines $r: 2x - y - 6 = 0$ and $s: x + y - 6 = 0$.

1. 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.
2. 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.
3. 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 the exact coordinates of the three vertices of the park.

Discuss the relative position of the lines $r: kx + y - 3 = 0$ and $s: 4x + ky - 6 = 0$ depending on the possible values of the parameter $k$.

A robot vacuum cleaner moves along the path described by the line $r: 3x + 4y - 24 = 0$. Upon encountering an obstacle at the exact point of intersection with the y-axis (axis of ordinates), it turns $90^{\circ}$ and continues straight ahead. Find the general equation of the new path and the point where it will intersect the x-axis (axis of abscissas).

Find the coordinates of the reflection of point $A(2, 5)$ in the line $r: x - 2y + 3 = 0$.

The sides of a quadrilateral-shaped plot of land lie on the lines $r: 2x + y - 5 = 0$ and $s: x - 2y + 5 = 0$. Given that one of the vertices of the plot is the point $P(4, 2)$ and that the enclosure is a perfect rectangle, find the general equations of the lines containing the other two sides.

Calculate the area of the triangle formed by the line $r: 5x + 2y - 20 = 0$ and the two coordinate axes.

Given the lines $r: 2x - y + 4 = 0$ and $s: x + y - 1 = 0$, find the explicit equation of a line parallel to $r$ that passes through the point of intersection of $s$ with the x-axis.

A ray of light follows the path of the line $r: x - y + 2 = 0$ and strikes a flat mirror located along the x-axis. Given that the angle of incidence equals the angle of reflection relative to the normal, find the general equation of the line describing the reflected ray.

The National Transport Authority in Ireland offers two monthly pass options for traveling on an intercity line:

• Frequent Option: a fixed fee of $15$ € per month plus $1.20$ € for each trip made.

• Occasional Option: no fixed fee, with a cost of $2.20$ € for each trip made.

1. a) Model the total monthly cost ($y$, in euros) according to the number of trips made per month ($x$) for each option, expressing both relationships as general equations of lines ($Ax + By + C = 0$).
2. b) Analytically justify the relative position of both lines from the ratio of their coefficients. Find the exact coordinates of the intersection point and interpret their practical meaning for a user when choosing which pass to buy.
3. c) The management entity plans to replace the Occasional Option with a new fare whose associated line is strictly parallel to that of the Frequent Option. What exact price per trip should it set for that new fare to achieve this? Would it be possible for both options to cost the same for some number of trips in that case? Justify your answer based on the geometric properties of parallel lines.

Consider the lines $r: kx + 3y - 6 = 0$ and $s: 3x + ky - (k + 3) = 0$, where $k$ is a real parameter:

1. a) Discuss the relative position of both lines according to the different values of $k$, clearly distinguishing through the proportionality of the coefficients when they are intersecting, parallel, or coincident.
2. b) For the particular case $k = 1$, the lines intersect. Calculate the exact coordinates of their intersection point and verify algebraically that this point belongs to both lines.
3. c) A classmate makes the following claim: «If two lines have the same direction, they are necessarily the same line». Relying on the conclusions obtained in section a) for the value $k = -3$, refute their claim with reasoning, explaining the algebraic and geometric difference between parallel and coincident lines.