Exercises: Equations of a line in the plane

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

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

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

Write the point-slope equation of the line that passes 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 Cartesian equation $\frac{x-2}{2} = \frac{y+1}{-2}$.

Find the explicit equation of the line that passes through the origin and has a slope of $m = -\frac{3}{5}$.

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

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

A surveillance drone moves in a straight line and at a constant speed from point $A(10, 20)$ to point $B(50, 80)$, with the coordinates measured in feet. Write the parametric equations of its path and indicate what its coordinates will be when it has traveled 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 that relates 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 Cartesian form.

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

Find the general equation of the line that passes through the point $A(-4, 6)$ and is parallel to the line joining 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 miles). Determine at which exact points the pipe crosses the coordinate axes.

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

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 $A(2, 5)$, strikes a flat mirror located along the x-axis, reflects and passes through the point $B(8, 3)$. Given 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 point of bounce on the mirror.

The sides of a triangle are contained in 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 decide, 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 the 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, given that $a = 2b$ and that both intercepts are positive.

Two airplanes move in the plane of a radar. Airplane $A$ follows the path $x = 2 + 3t$, $y = 1 + 4t$, and airplane $B$ follows the path $x = 10 - t$, $y = -2 + 5t$ (where $t$ is the time in minutes and the coordinates are in miles). Determine the exact point where their paths cross and find out if the airplanes will collide (that is, if they pass through that point at the same instant $t$).

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

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

Consider the pencil of lines determined 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 of the pencil has a slope of $\frac{3}{5}$.