The circle

Write the equation of the circle describing the coverage area of a Wi-Fi antenna, knowing that the antenna is located at point $(2, -3)$ on a Cartesian plane (measured in kilometres) and has a maximum range of $5$ km.

Determine the coordinates of the centre and the length of the radius of the circular area watered by an agricultural sprinkler whose boundary is given by the equation $(x-4)^2+(y+1)^2=36$.

Find the equation of the circle representing the sweep of a ship's radar. It is known that the radar is positioned at coordinates $(-1, 4)$ and that at the exact edge of its screen it detects a buoy located at point $(2, 0)$.

Write the equation of the circle bounding a roundabout, given that the endpoints of one of its diameters (where two opposite pedestrian crossings are located) correspond to the points $(1, 2)$ and $(7, -6)$.

Write the equation of the circle describing the chassis of a robot vacuum cleaner of radius $15$ cm when it is wedged exactly into the corner of a room. Consider that the walls coincide with the positive semi-axes of coordinates (quadrant I) and the robot is tangent to both walls.

Find the centre and radius of the circle describing the outer edge of a gear in a computer-aided design blueprint, whose general equation is $x^2+y^2-6x+8y=0$.

Determine whether the equation $2x^2+2y^2+4x-8y+20=0$ mathematically represents a valid circle in a video game physics engine. Justify your answer by calculating its centre and the value of its radius squared.

Lucas needs to calculate the center and the radius of the circle given by the general equation $x^2 + y^2 - 6x + 4y - 12 = 0$. On his exam, he writes the following:

"The center is $C(-6, 4)$ because these are directly the coefficients accompanying $x$ and $y$. For the radius, I substitute into the formula and obtain $r = \sqrt{(-6)^2 + 4^2 - 12} = \sqrt{36 + 16 - 12} = \sqrt{40} \approx 6.32$".

Identify and explain in detail the two conceptual mistakes Lucas made when deducing the center and the radius from the general equation. Then, show the correct reasoning and calculate the exact coordinates of the center $C$ and the exact value of the radius $r$.

Consider the family of curves given by the equation $x^2 + y^2 - 4x + 6y + k = 0$, where $k$ is a real number.

1. a) Determine with reasoned steps the value that the parameter $k$ must take so that the equation represents a circle of radius exactly equal to $5$ units, and write its corresponding standard form equation.
2. b) Justify what geometric figure or object this equation represents if $k = 13$ is taken, and what happens if any value $k > 13$ is chosen.

Find the equation of the circle that reconstructs the rim of an ancient ceramic vessel found in an archaeological excavation. Restorers have measured three intact points on the rim, which in a coordinate axis system correspond to $(0, 0)$, $(4, 0)$ and $(0, 3)$.

Find the equation of the circumscribed circle of the triangular park bounded by the avenues whose equations are $x=0$, $y=0$ and $3x+4y-12=0$.

A seismograph detects that the epicentre of an earthquake is located at the origin of coordinates and that the seismic wave has reached an exact radius of $10$ km. Check analytically whether a weather station located at point $(6, -8)$ felt the earthquake or if it is on the exact boundary of the wave.

Study analytically the relative position between the orbit of an artificial satellite, modelled by the circle $x^2+y^2=4$, and the straight-line trajectory of an asteroid given by the equation $y=x+2$. Determine the points of intersection if they exist.

Calculate the equation of the tangent line to the circular trajectory traced by a hammer thrower at the exact moment they release the wire. The trajectory of the hammer is given by the circle $x^2+y^2=25$ and the athlete releases the wire when the ball passes through the point $(3, 4)$.

Write the equation of the circle representing a pulley whose axis of rotation is fixed at the point $(3, -2)$ and which is in tangential contact with a drive belt whose line of action is the line $4x-3y+2=0$.

Calculate the length of the free surface of the water inside a circular pipe of equation $x^2+y^2=100$. It is known that the water level has stabilised, forming a horizontal chord along the line $y=6$.

A fishing boat sails in a straight line along the path given by the equation $3x + 4y - 24 = 0$ (with coordinates expressed in nautical miles). A maritime beacon of the Irish Coast Guard emits a storm warning with a circular coverage area bounded by a radius of $r = 3.5$ nautical miles and centered at the point $C(1, 1)$.

1. a) Before calculating anything, make a reasoned estimate, mentally or with a sketch: taking into account the position of the center and the points where the line meets the Cartesian axes, do you think the fishing boat will cross the warning zone, just graze it tangentially, or pass by without entering it? Justify your conjecture.

1. b) Check your estimate analytically by calculating the distance from the center $C$ to the line of the path and comparing it with the radius to conclusively determine the relative position. Explain why, when the distance and the radius are very close values, a simple visual representation is not enough and algebraic calculation is essential.

Find the equations of the two tangent lines drawn from a spotlight located at the point $(0, 4)$ to a cylindrical fuel tank whose base occupies the region bounded by the circle $x^2+y^2=4$. These lines represent the boundaries of the shadows cast on the ground.

The design of a road junction requires drawing two straight roads that are tangent to a central circular plaza whose equation is $x^2+y^2-6x+8y=0$. Furthermore, for urban planning reasons, these roads must be parallel to a main avenue with equation $3x-4y=0$. Find the equations of both tangent roads.

A drone performs an evasive manoeuvre following a straight-line trajectory given by the family of lines $y=2x+k$. The restricted airspace around a radar is defined by the circle $x^2+y^2=5$. Determine the values of the parameter $k$ for which the drone grazes the restricted airspace without penetrating it (that is, the line is tangent to the circle).

To draw the tangent lines to the circle $x^2 + y^2 - 4x - 2y + 1 = 0$ from the external point $P(6, 4)$, two different analytical methods can be proposed:

• Method 1 (purely algebraic): Set up the family of lines passing through $P$ using its slope $m$, substitute $y$ into the circle's equation, and require that the discriminant of the quadratic equation in $x$ be zero ($\Delta = 0$).

• Method 2 (metric-geometric): Express the family of lines in general form $mx - y + (4 - 6m) = 0$ and require that the Euclidean distance from the center of the circle to the line equals the radius exactly ($d(C, r) = R$).

1. a) Analyze the advantages and disadvantages of both methods (computational workload, risk of complex operations, and handling of a possible vertical tangent) and justify which one is more efficient in this case.
2. b) Solve the problem applying the method you selected as optimal, find the equations of the two tangent lines, and justify why neither of the tangents starting from $P$ is a vertical line.

Find the equation of the radical axis of the two circles modelling the area of influence of two radio antennae: $x^2+y^2-4x+2y-4=0$ and $x^2+y^2+6x-2y+6=0$.

Study the relative position between the wavefronts generated by two drops falling into a pond, modelled by the circles $C_1: x^2+y^2=9$ and $C_2: (x-4)^2+y^2=4$.

Determine the equation of the circle defining the outer lane of a running track, given that it is concentric to the inner lane circle $x^2+y^2-10x+4y+13=0$ and passes through the starting position located at $(1, 1)$.

The 112 emergency service of a county in Ireland evaluates communication coverage for wildfires using two repeater antennas. On the command map (with a grid scaled in tens of kilometers, where $1$ unit $= 10$ km), the North antenna is located at $A(-3, 2)$ with a coverage radius of $50$ km ($5$ units), and the South antenna at $B(5, -4)$ with the same range of $50$ km.

1. a) Algebraically model the coverage boundary of both antennas by writing their respective general equations.
2. b) Study the relative position between the two circular coverage areas from the distance between their centers and deduce whether there is territorial overlap or if they are tangent.
3. c) The advanced command post is located in the town with coordinates $P(1, -1)$. Determine whether this town has coverage from either antenna, from both, or if it remains cut off in a blind spot, interpreting the geometric result in the context of emergencies.

Calculate the equation of the circle outlining the main structure of a Ferris wheel. It is known that the central axis of rotation rests on an inclined pillar represented by the line $x+y-4=0$ and that the cabins pass through the boarding platforms located at the points $(2, 0)$ and $(0, 2)$.

In an acoustics experiment, we seek the locus of all points in the plane where the sound emitted by a loudspeaker at $A(6, 0)$ is perceived with a quarter of the intensity of the sound emitted by another identical loudspeaker at the origin $O(0, 0)$. This is equivalent to finding the points $P(x, y)$ whose distance to the point $A$ is twice their distance to the point $O$. Prove that this locus is a circle and find its centre and its radius.

The GPS system of a mobile phone receives signals from two telecommunications towers. The first tower is at the origin of coordinates and estimates that the mobile phone is $5$ km away. The second tower is at the point $(5, 0)$ and estimates that the mobile phone is $4$ km away. Find the exact coordinates of the two possible points where the mobile phone could be located by calculating the intersection of both circles.

In a network optimisation problem, it is required to find the locus of points in the plane such that the sum of the squares of their distances to two fixed servers located at $(-2, 0)$ and $(2, 0)$ is exactly equal to $20$. Prove analytically that the resulting figure is a circle and obtain its equation.

In an urban planning project, a circular roundabout is designed whose center must necessarily lie on the axis of a straight avenue modeled by the equation $r: x - 2y = 0$. In addition, the outer edge of the roundabout must be tangent to a tram line given by the line $s: 4x + 3y - 1 = 0$, and its path must pass through a historical monument located at the point $A(2, 3)$.

1. a) Parameterize the coordinates of the center in terms of a single variable, set up the equality between the distance from the center to point $A$ and the distance from the center to line $s$, and prove with reasoned steps that there are two theoretical circles satisfying all the given conditions, finding their corresponding centers and radii.
2. b) From the perspective of technical feasibility and the dimensions of a roundabout in an ordinary urban fabric, critically justify why one of the two theoretical solutions completely lacks practical sense and must be discarded. For the feasible solution, write its general equation and calculate the exact coordinates of the point of tangency with the tram line.