Exercises: Circle

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

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

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

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

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

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

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

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

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

Study analytically the relative position between the orbit of an artificial satellite, modeled by the circle $x^2+y^2=4$, and the rectilinear 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 cable. The trajectory of the hammer is given by the circle $x^2+y^2=25$ and the athlete releases the cable when the ball passes through the point $(3, 4)$.

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

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

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

Study the relative position between the wavefronts generated by two drops falling into a pond, modeled 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 that defines the outer lane of a running track, knowing that it is concentric to the circle of the inner lane $x^2+y^2-10x+4y+13=0$ and that it passes through the starting position located at $(1, 1)$.

Find the equations of the two tangent lines drawn from a lighthouse 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 limits of the shadows cast on the ground.

Calculate the equation of the circle that outlines 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 are looking for the locus of all points in the plane where the sound emitted by a loudspeaker located 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 center and its radius.

The design of a road junction requires drawing two straight roads that are tangent to a central circular square 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 maneuver following a rectilinear 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).

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

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

In a network optimization problem, it is required to find the locus of the 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.