Exercises: Equations of a line in the plane

Write the vector equation and parametric equations of the line passing through the point $A(3, -2)$ with direction vector $\vec{v} = (-1, 4)$.

Given the line in parametric equations $x = 2 - 3t$, $y = 1 + 5t$, express it in continuous form and in general form.

Determine the gradient and the y-intercept of the line whose general equation is $4x - 2y + 8 = 0$.

Write the point-gradient equation of the line passing through the points $P(-1, 5)$ and $Q(2, -1)$.

Check whether the points $M(4, -3)$ and $N(1, 2)$ belong to the line with continuous equation $\frac{x-2}{2} = \frac{y+1}{-2}$.

Find the explicit equation of the line passing through the origin of coordinates and having a gradient of $m = -\frac{3}{5}$.

Write the implicit equations of the horizontal line passing through the point $H(-7, 4)$ and the vertical line passing through the point $V(3, -9)$.

Given the implicit equation $5x + 7y - 11 = 0$, state a direction vector, a normal vector, and a point through which the line passes.

In a Mathematics exam, Cian must write the continuous equation of the line that passes through the point $A(-3, 5)$ and has direction vector $\vec{v} = (2, -4)$. On his sheet, he writes:

When grading it, his teacher points out that the answer contains a widespread conceptual error.

1. a) Identify precisely the two sign mistakes Cian made in the numerators of the expression.
2. b) Explain the origin of this error by deriving the continuous equation step by step from the parametric equations: first write $x = a_1 + t\times v_1$ and $y = a_2 + t\times v_2$ with the given data and solve for the parameter $t$ in each one to show why the signs must be the opposite of what Cian wrote.
3. c) Write the correct continuous equation and, from it, obtain the general equation of the line by simplifying the integer coefficients as much as possible.

Design the rectilinear path of a transport robot in a logistics center meeting the following specifications:

1. a) Write parametric equations of a line that passes through the point $P(4, -1)$ and whose slope is exactly $m = -\frac{2}{3}$.
2. b) From the previous data, directly write the point-slope equation of the line using point $P$ as reference.
3. c) A classmate claims that another valid answer for the parametric equations of the same line is $x = 1 - 6k$, $y = 1 + 4k$ ($k \in \mathbb{R}$). Check whether the point she used as a base belongs to the line and whether her direction vector is collinear with yours. Conclude by arguing why the parametric equations of a line are not unique and which elements can vary while geometrically representing the same line.

A surveillance drone moves in a straight line at a constant speed from point $A(10, 20)$ to point $B(50, 80)$, with the coordinates measured in metres. Write the parametric equations of its trajectory and state its coordinates when it has travelled exactly half the distance.

The production cost of an electronic component follows a linear model. Manufacturing $100$ units costs $450$ €, and manufacturing $250$ units costs $825$ €. Determine the explicit equation relating the cost $y$ to the number of units $x$, and calculate the fixed production cost (the cost when $0$ units are manufactured).

Determine the value of the parameter $k$ so that the line with equation $3x - ky + 5 = 0$ passes through the point $P(-2, 1)$. Once $k$ is found, write the equation of the line in continuous form.

In the design of a suspension bridge, one of the steel tension cables must connect the anchor point $P(0, 15)$ to the base $Q(40, 0)$ (coordinates in metres). Find the general equation of the line defining the cable and determine its height when the horizontal distance from the $Y$-axis is $16$ metres.

Find the general equation of the line passing through the point $A(-4, 6)$ and parallel to the line connecting the points $B(1, 1)$ and $C(3, -5)$.

An underground pipe follows the path given by the parametric equations $x = -5 + 2t$, $y = 12 - 3t$ (in km). Determine the exact points at which the pipe crosses the coordinate axes.

Calculate the area of the triangle formed by the line $2x + 5y - 20 = 0$ with the Cartesian coordinate axes.

Express the line $y = -\frac{1}{4}x + 2$ in vector form and find three distinct points that belong to it, where at least one has a negative x-coordinate.

Consider the line $r$ that passes through the points $A(-2, 7)$ and $B(4, -5)$.

1. a) Before calculating any analytical equation:

• Observe how the coordinates change when going from $A$ to $B$ and estimate the sign of the slope (does the line rise or fall from left to right?).

• Knowing that point $A$ has $x = -2$ with $y = 7$ and point $B$ has $x = 4$ with $y = -5$, reasonably estimate whether the $y$-intercept $n$ (the intersection with the $Y$-axis, where $x = 0$) will be positive or negative, and whether its value will be closer to $7$ or to $-5$.

1. b) Write the continuous equation of the line $r$ and transform it through neat algebraic steps until obtaining its explicit equation $y = mx + n$.
2. c) From the explicit equation, calculate the exact coordinates of the points of intersection with the $y$-axis ($x = 0$) and with the $x$-axis ($y = 0$). Check whether the value of the $y$-intercept matches your initial estimate and geometrically justify why that value is at that height.

We want to find the general equation of the line $s$ that passes through the point $P(-3, 2)$ and is perpendicular to the line $r: 2x - 5y + 11 = 0$. Two students propose different strategies:

• Seán's method: Solve for $y$ in the line $r$ to find its slope $m_r$; calculate the perpendicular slope $m_s = -\frac{1}{m_r}$; write the equation in point-slope form with $P(-3, 2)$ and, finally, clear denominators and rearrange terms to reach the general equation.

• Aoife's method: Directly obtain from the coefficients of $r$ its normal vector $\vec{n}_r = (2, -5)$; deduce that this vector is a direction vector of the perpendicular line $s$; extract from it a normal vector for $s$, $\vec{n}_s = (5, 2)$; and set up the equation $5x + 2y + C = 0$, finding $C$ by substituting the coordinates of $P$.

1. a) Compare both approaches by analyzing their advantages and disadvantages: number of steps, appearance of intermediate fractions, and likelihood of making operational sign errors.
2. b) Justify which of the two methods is more efficient and direct for this exercise.
3. c) Apply the method you selected as preferable to determine the general equation of $s$.

Determine the values of the parameters $a$ and $b$ so that the line given in parametric form $x = a + 2t$, $y = 3 - bt$ represents the same line as the general equation $4x + 3y - 10 = 0$.

A laser beam starts from the point $A(2, 5)$, hits a flat mirror situated along the x-axis, reflects, and passes through the point $B(8, 3)$. Knowing that the angle of incidence is equal to the angle of reflection, find the equations of the lines containing the incident beam and the reflected beam, and the exact bounce point on the mirror.

The sides of a triangle lie on the lines $r: x - 2y + 4 = 0$, $s: 3x + y - 9 = 0$ and $t: x + 5y + 4 = 0$. Calculate the coordinates of its three vertices and determine, by calculating the lengths of its sides, what type of triangle it is.

Prove that if a line intersects the coordinate axes at the points $(a, 0)$ and $(0, b)$, with $a \neq 0$ and $b \neq 0$, its equation can be written in intercept form $\frac{x}{a} + \frac{y}{b} = 1$. Use this result to find the equation of a line that forms a triangle of area $12 \text{ u}^2$ with the Cartesian axes, knowing that $a = 2b$ and that both intercepts are positive.

Two aeroplanes move in a radar plane. Aeroplane $A$ follows the trajectory $x = 2 + 3t$, $y = 1 + 4t$, and aeroplane $B$ follows the trajectory $x = 10 - t$, $y = -2 + 5t$ (where $t$ is the time in minutes and coordinates are in km). Determine the exact point where their trajectories cross and find out if the aeroplanes collide (that is, if they pass through that point at the same instant $t$).

Find the equation of the line that passes through the intersection point of the lines $r: 2x - y + 5 = 0$ and $s: x + 3y - 1 = 0$, and forms an isosceles triangle with the negative coordinate semi-axes.

Given the point $P(3, 4)$ and the line $r: 2x + y - 5 = 0$, find the coordinates of a point $Q$ situated on the y-axis such that the line connecting $P$ and $Q$ has the same y-intercept as the line $r$.

Consider the pencil of lines defined by the equation $(2 + k)x + (1 - 2k)y + (3k - 4) = 0$. Find the centre of the pencil (the point through which all the lines pass regardless of the value of $k$) and determine the value of $k$ for which the line in the pencil has a gradient of $\frac{3}{5}$.

In a meteorological study by Met Éireann in the Wicklow Mountains National Park (Wicklow), the variation of temperature with altitude is analyzed during an anticyclonic winter day. At the Glendalough station (altitude $x_1 = 2100\text{ m}$), a temperature of $y_1 = 8.4\ ^\circ\text{C}$ is measured at noon, while at the station near the summit of Lugnaquilla (altitude $x_2 = 3100\text{ m}$), the thermometer simultaneously reads $y_2 = 1.9\ ^\circ\text{C}$. A linear relationship between altitude $x$ (in meters) and temperature $y$ (in $^\circ\text{C}$) is assumed.

1. a) Write the continuous equation that models this linear relationship using Glendalough as the reference point and a direction vector whose components reflect the temperature variation per $1000\text{ m}$ of ascent.
2. b) Transform this equation into explicit form $y = mx + n$. Rigorously interpret the physical meaning of the slope $m$ (vertical lapse rate) and of the $y$-intercept $n$. Does the numerical value of $n$ make physical sense when compared to a plausible sea-level temperature ($x = 0$) on the Irish east coast?
3. c) Use the linear model to calculate the altitude of the snow level (freezing level: altitude where the temperature drops to $0\ ^\circ\text{C}$). From that data, justify whether a mountaineer reaching the summit of Carrauntoohil ($3479\text{ m}$) will record positive or below-zero temperatures at that same moment.

A line $r$ with a negative slope passes through the point $A(2, 3)$ and intersects the positive coordinate axes at points $P(p, 0)$ and $Q(0, q)$, forming with the origin of coordinates a right triangle located in the first quadrant whose area is exactly $12\text{ u}^2$.

1. a) Deduce step by step that the continuous equation of the line passing through $(p, 0)$ and $(0, q)$ can be rewritten in the intercept form $\frac{x}{p} + \frac{y}{q} = 1$.
2. b) Knowing that the point $A(2, 3)$ belongs to the line and that the area of the triangle is $\frac{p\times q}{2} = 12$, set up a system of two algebraic equations with unknowns $p$ and $q$.
3. c) Solve the system and show that there is only one pair of real values $(p, q)$ that satisfies both conditions simultaneously. With these values, write the general equation of the line.
4. d) Analyze algebraically why the discriminant of the quadratic equation that arises when solving the system turns out to be exactly zero for an area of $12\text{ u}^2$, and interpret geometrically what would happen to the existence of lines passing through $A(2,3)$ if we had required the area of the triangle to be smaller, for example $10\text{ u}^2$.