Equations of a Line in the Plane

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

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

The production cost of an electronic component follows a linear model. Manufacturing $100$ units costs $27\,000$ pesos, and manufacturing $250$ units costs $49\,500$ 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).

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

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.

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

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.

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.

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

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.

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 coordinate axes.

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 $4x + 3y - 10 = 0$.

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

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

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.

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 $C(3, -5)$.

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 $t$).

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.

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.

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.

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 regardless of the value of $k$) and determine the value of $k$ for which the line in the pencil has a slope of $\frac{3}{5}$.