Exercises: Equation of a line

Given the general equation of the line $r: 3x - 5y + 7 = 0$, a student quickly claims that its direction vector is $\vec{v} = (3, -5)$. Identify the mistake made when extracting information from the general equation and determine a correct direction vector and normal vector for this line.

Write the vector equation and parametric equations of the line that passes through point $A(2, -5)$ and has direction vector $\vec{v} = (3, 4)$.

Obtain the symmetric equation of the line that passes through point $P(-1, 6)$ and has the same direction as vector $\vec{u} = (-3, 2)$.

Write the explicit equation of the line that passes through point $B(4, 1)$ and has gradient $m = -\frac{1}{2}$.

Express in general form the line given by the following symmetric equation:

$\frac{x-3}{2} = \frac{y+1}{-5}$

Verify analytically whether points $C(3, 2)$ and $D(-1, 4)$ lie on the line with equation $2x + y - 8 = 0$.

Given the line with equation $3x - 4y + 12 = 0$, find the value of its gradient and its y-intercept.

Determine the coordinates of two different points lying on the line whose parametric equations are:

$x = 1 - t$

$y = 2 + 3t$

Identify the coordinates of a point and the components of a direction vector of the line given by the equation:

$\frac{x+4}{-3} = \frac{y-2}{7}$

Elena has to calculate the general equation of the line that passes through the point $A(3, -1)$ and has direction vector $\vec{v} = (-2, 5)$. In her notebook, she works through the following steps:

1. She writes the symmetric form: $\dfrac{x - 3}{-2} = \dfrac{y + 1}{5}$.

2. She cross-multiplies: $5 \cdot (x - 3) = -2 \cdot (y + 1) \implies 5x - 15 = -2y - 2$.

3. She moves everything to the first side: $5x + 2y - 17 = 0$.

Then, to check whether point $B(-1, 9)$ belongs to the line, Elena substitutes the coordinates into her result and concludes that it does belong because $5 \cdot (-1) + 2 \cdot 9 - 17 = -4 \ne 0$.

1. a) Find and explain the two mistakes made by Elena (an algebraic one in step 3 when rearranging terms, and a conceptual one in the final interpretation regarding point $B$).
2. b) Write the correct general equation and check analytically whether point $B(-1, 9)$ truly belongs to the line.

Design with justification a line in symmetric form that simultaneously satisfies these three conditions:

1. It intersects the y-axis ($OY$) at the point $(0, -3)$.

2. Its direction vector $\vec{v} = (v_1, v_2)$ consists of positive integers and its slope is less than $1$.

3. It passes through the point $P(6, 1)$.

1. a) Deduce the components of a valid direction vector from the given points and justify why its slope satisfies being less than $1$.
2. b) Write the resulting symmetric equation.
3. c) Justify why there are infinitely many possible direction vectors to express this same line, but only one whose components are coprime positive integers.

A maritime surveillance drone travels in a straight line at constant speed. At 10:00 it is detected at point $P(2, 5)$ and at 10:15 at point $Q(8, -1)$ (Cartesian coordinates in km). Find the explicit equation of the trajectory described by the drone.

In the planning of a new neighbourhood, the main street follows the line $r: 2x - 5y + 10 = 0$. A secondary street parallel to the main street is to be laid out, passing through the central square located at $A(4, -3)$. Determine the general equation of this new street.

A water pipe runs along the line $s: x + 3y - 6 = 0$. It is necessary to connect a house located at $C(-2, 5)$ by means of a pipe perpendicular to the main pipe. Find the parametric equations of the connecting pipe.

Two observation satellites follow straight-line trajectories described by equations $r: 3x + y - 7 = 0$ and $s: x - 2y + 4 = 0$. Calculate the exact coordinates of the point where their routes intersect.

The production cost $y$ (in hundreds of euros) of $x$ units of an electronic component follows a linear model. If we know that the production line passes through point $(10, 15)$ and has direction vector $\vec{v} = (2, 1)$, find the general equation of the model and the fixed cost (the value of $y$ when $x=0$).

Determine the value of the parameter $k$ such that the line with equation $(k-1)x + 2y - 5 = 0$ passes through point $P(3, -1)$.

Write the general equations of the four lines that form the boundary of a rectangular tennis court, knowing that two of its adjacent sides lie on the coordinate axes and the vertex opposite the origin is located at $V(24, 11)$.

Calculate the value of $a$ such that lines $r: ax + 3y - 2 = 0$ and $s: 4x - 6y + 5 = 0$ are perpendicular to each other.

According to data from AEMET, under standard atmospheric conditions air temperature decreases roughly linearly with altitude in the troposphere. During a spring day in Sierra Nevada (Granada), the following data are recorded at 14:00:

• Station 1 (elevation $1000\text{ m}$, $x = 1\text{ km}$): $14\text{ }^{\circ}\text{C}$.

• Station 2 (elevation $3000\text{ m}$, $x = 3\text{ km}$): $2\text{ }^{\circ}\text{C}$.

1. a) Determine the direction vector of the line relating altitude $x$ (in km) to temperature $y$ (in $^{\circ}\text{C}$) and write its slope-intercept equation $y = mx + n$.
2. b) Interpret the physical meaning of the slope $m$ and the y-intercept $n$ in this meteorological context.
3. c) A group of mountaineers reaches the summit of Mulhacén peak ($3480\text{ m}$, that is, $x = 3,48\text{ km}$). Use your model to estimate the temperature at the summit at that time and reason whether there is a risk of frost (temperature $\le 0\text{ }^{\circ}\text{C}$).

A group of hikers walks along a straight section in Picos de Europa National Park. On their topographic map (with a grid in kilometres), the trail follows a straight line connecting the refuge $R(1, 2)$ with the viewpoint $M(7, 5)$. Off the trail, at point $F(4, 4)$, there is a natural spring.

1. a) Estimate: Notice that the x-coordinate $x = 4$ of the spring is exactly halfway horizontally between $x = 1$ and $x = 7$. Without calculating the equation of the line, estimate with reasoning whether the path will pass north (above, $y > 4$) or south (below, $y < 4$) of the spring.
2. b) Verify: Obtain the symmetric equation and the general equation of the line defining the trail $RM$.
3. c) Calculate the exact y-coordinate $y$ of the path when the hikers are at $x = 4$. Compare this value with your previous estimate and state the exact vertical distance from the spring at which they will pass.

A border wall is to be built to divide equally the sphere of influence of two telecommunications antennas located at $A(-4, 2)$ and $B(6, -2)$. The wall must coincide exactly with the perpendicular bisector of segment $AB$. Find the general equation of the line defining this wall.

A laser light beam starts from point $P(-3, 5)$ and reflects off a flat mirror located along the x-axis. After bouncing, the reflected beam passes through point $Q(4, 2)$.

• Find the explicit equations of the lines containing the incident beam and the reflected beam.

• Calculate the exact coordinates of the point of impact on the mirror.

The lines enclosing a triangular urban park are $r: x - y + 4 = 0$, $s: 2x + y - 7 = 0$ and $t: x + 2y + 1 = 0$. Calculate the coordinates of the three vertices of the park.

Find the general equation of the line that passes through the point of intersection of lines $r: 2x - y + 3 = 0$ and $s: x + 4y - 12 = 0$, and also forms an angle of $45^\circ$ with the positive $X$-axis.

A triangular solar panel has its vertices located at coordinates $A(1, 1)$, $B(5, 3)$ and $C(2, 6)$. Calculate the general equations of the three altitudes of the triangle and analytically verify that they intersect at a single point (the orthocentre).

Determine the area of the triangular region bounded by the coordinate axes and the line passing through point $M(3, 4)$ and perpendicular to vector $\vec{n} = (2, 5)$.

Discuss the relative position of lines $r: mx + 2y - 4 = 0$ and $s: 3x + (m-1)y - 6 = 0$ depending on the possible values of the parameter $m$.

In a game of billiards, a ball is struck from point $A(2, 1)$ towards the top cushion of the table, which lies along line $y = 5$. After bouncing off the cushion, the ball follows a straight line trajectory and hits another ball located at point $B(8, 3)$. Calculate the exact coordinates of the point on the cushion where the ball bounced, knowing that in an elastic collision the angle of incidence equals the angle of reflection.

The Tarifa Maritime Rescue centre monitors a drifting boat in the Strait of Gibraltar. Its initial position at 08:00 is $P(3, 8)$ and the currents move it along a straight line. Its Cartesian coordinates (in nautical miles) evolve according to the time parameter $t$ (in hours) via the parametric equations:

Meanwhile, a rescue patrol boat patrols along a fixed line whose trajectory is given by the general equation $3x + 2y - 37 = 0$.

To determine whether their trajectories intersect and at what moment, two methods are proposed:

• Method 1: Convert the drifting boat's parametric equations to their general equation and solve a system of two linear equations in two variables ($x$ and $y$).

• Method 2: Directly substitute the parametric expressions $x(t)$ and $y(t)$ into the patrol boat's general equation and solve for time $t$.

1. a) Critically justify which of the two methods is faster and more direct in this particular case.
2. b) Apply the chosen method to calculate the value of the parameter $t$ and the exact coordinates $(x, y)$ of the intersection point.
3. c) Interpret the result physically: at what exact time does the encounter occur? What would it have meant if the value obtained for $t$ had been negative?

At the PLAZA logistics platform in Zaragoza, two freight train convoys travel along straight tracks represented on a coordinate map (in km):

• Convoy A departs from the logistics terminal located at $A(-6, -1)$ and its direction follows the direction vector $\vec{v}_A = (4, 3)$.

• The track of convoy B is given by the general equation $6x - 8y + 53 = 0$.

1. a) Reasonably analyse the relative position of both tracks (are they intersecting, parallel, or coincident?). To justify your answer, compare their slopes or direction vectors and check if they share any common point.
2. b) To facilitate maintenance, a straight service track is planned perpendicular to the trajectory of convoy A, starting precisely from terminal $A(-6, -1)$. Find the slope-intercept equation of this service track.
3. c) A third convoy breaks down and remains stopped at point $C(2, 5)$. Determine analytically whether it lies on the trajectory of convoy A. If so, deduce how many times the direction vector $\vec{v}_A$ had to be shifted from the origin $A$ to reach the position of the broken-down convoy.