Equations of a Line in the Plane
A surveillance drone flies over Mexico City in a straight line, and the control system describes it with an equation; on the UNAM or IPN entrance exam, that same line appears written in a different form. They are not different lines: it is the same line in different "disguises." If you understand where each form comes from, going from one to another is mechanical.
The line as a point and a direction: vector equation
Write the vector equation and parametric equations of the line passing through the point $A(3, -2)$ with direction vector $\vec{v} = (-1, 4)$.
Write the vector equation and parametric equations of the line passing through the point $A(3, -2)$ with direction vector $\vec{v} = (-1, 4)$.
Separating into $x$ and $y$: parametric equations
Axis intercepts from the parametric equations
A point is on the $X$-axis when $y=0$ and on the $Y$-axis when $x=0$. With parametric equations, you impose that condition, solve for $t$, and substitute it into the other coordinate.
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 meters. Write the…
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 meters. Write the parametric equations of its trajectory and state its coordinates when it has traveled exactly half the distance.
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…
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.
Eliminating the parameter: continuous equation
Check whether the points $M(4, -3)$ and $N(1, 2)$ belong to the line with continuous equation $\frac{x-2}{2} = \frac{y+1}{-2}$.
Check whether the points $M(4, -3)$ and $N(1, 2)$ belong to the line with continuous equation $\frac{x-2}{2} = \frac{y+1}{-2}$.
Given the line in parametric equations $x = 2 - 3t$, $y = 1 + 5t$, express it in continuous form and in general form.
Given the line in parametric equations $x = 2 - 3t$, $y = 1 + 5t$, express it in continuous form and in general form.
Clearing denominators: general equation
Write the implicit equations of the horizontal line passing through the point $H(-7, 4)$ and the vertical line passing through the point $V(3, -9)$.
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.
Given the implicit equation $5x + 7y - 11 = 0$, state a direction vector, a normal vector, and a point through which the line passes.
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 $C(3, -5)$.
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 $C(3, -5)$.
Slope-intercept and point-slope forms
Before solving for anything, let's think about direct proportionality. If two variables are directly proportional, their ratio is constant: $\dfrac{y}{x}=m$, that is, $y=mx$. Its graph is a line that passes through the origin, and $m$ is the slope: how much $y$ increases for each unit that $x$ increases.
If there is also a fixed amount from the start, the line shifts $b$ units up and becomes $y=mx+b$. For example, if a torta costs $\$45$ and delivery costs a flat $\$20$, the total for $x$ tortas is $y=45x+20$: the part $45x$ is proportional and the $\$20$ is the shift.
Determine the gradient and the y-intercept of the line whose general equation is $4x - 2y + 8 = 0$.
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 $Q(2, -1)$.
Write the point-gradient equation of the line passing through the points $P(-1, 5)$ and $Q(2, -1)$.
Find the explicit equation of the line passing through the origin of coordinates and having a gradient of $m = -\frac{3}{5}$.
Find the explicit equation of the line passing through the origin of coordinates and having a gradient of $m = -\frac{3}{5}$.
The production cost of an electronic component follows a linear model. Manufacturing $100$ units costs $9\,000$ pesos, and manufacturing $250$ units costs $16\,500$…
The production cost of an electronic component follows a linear model. Manufacturing $100$ units costs $9\,000$ pesos, and manufacturing $250$ units costs $16\,500$ pesos. 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).
Intercept form
Calculate the area of the triangle formed by the line $2x + 5y - 20 = 0$ with the Cartesian coordinate axes.
Calculate the area of the triangle formed by the line $2x + 5y - 20 = 0$ with the Cartesian coordinate axes.
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…
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.
Converting from one form to another
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…
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 meters). Find the…
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 meters). Find the general equation of the line defining the cable and determine its height when the horizontal distance from the $Y$-axis is $16$ meters.
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.
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.
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…
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 $4x + 3y - 10 = 0$.
Problems with lines
Here you combine forms and other ideas: intersection of lines (solving a system), distance between points, reflections, parallelism (same slope), and pencils of lines.
Distance between two points
To classify a triangle by its sides you need to measure them. The distance between $P(x_1,y_1)$ and $Q(x_2,y_2)$ comes from the Pythagorean theorem: segment $PQ$ is the hypotenuse of a right triangle whose legs measure $|x_2-x_1|$ and $|y_2-y_1|$.
Reflection of a point across the $X$-axis
A ray of light bouncing off a mirror satisfies the condition that the angle of incidence equals the angle of reflection. For a mirror along the $X$-axis there is a trick: the reflection of $B(x,y)$ is $B'(x,-y)$, and the reflected ray looks as if it comes in a straight line from $B'$. So the path $A\to$ bounce $\to B$ is equivalent to the line $AB'$, and the bounce point is where that line crosses the $X$-axis.
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…
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,…
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.
Two aeroplanes move in a radar plane. Aeroplane $A$ follows the trajectory $x = 2 + 3t$, $y = 1 + 4t$, and aeroplane $B$ follows the trajectory $x = 10 - t$,…
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 $t$).
Pencil of lines
A pencil of lines is the family of all the lines that pass through the same point, called the center of the pencil. If two lines $r: A_1x+B_1y+C_1=0$ and $s: A_2x+B_2y+C_2=0$ intersect at a point $P$, the pencil centered at $P$ is
$$(A_1x+B_1y+C_1)+k\,(A_2x+B_2y+C_2)=0, \qquad k\in\mathbb{R}$$Why? At $P$ both expressions equal $0$, so the sum equals $0$ for any $k$: all the lines pass through $P$. Each value of $k$ gives a different line of the pencil.
If you are given a pencil with $k$ mixed into the coefficients, split it into two parts: the one without $k$ and the one with $k$. Both parts are lines, and their intersection is the center.
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…
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…
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 center of the pencil (the point through which all the lines pass…
Consider the pencil of lines defined by the equation $(2 + k)x + (1 - 2k)y + (3k - 4) = 0$. Find the center 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 slope of $\frac{3}{5}$.
Go back to the beginning: the drone's path is written in parametric form, and that same line can be converted to general form or slope-intercept form; it is the same line in a different disguise.