The Circle
A Wi-Fi antenna on your block, a ship's radar, the ripple that spreads when a drop falls into a pond: all of them are described by the simplest of the conic sections. Can you tell with a single equation whether your cell phone is inside the antenna's coverage area? That is what you are going to work out here.
The Standard Equation
If the center is the origin, the equation becomes $x^2+y^2=r^2$.
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…
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 kilometers) and has a maximum range of $5$ km.
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…
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 $(x-4)^2+(y+1)^2=36$.
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…
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 $(2, 0)$.
Common Cases
A point $P$ is inside, on, or outside the circle depending on whether $d(P, C)$ is less than, equal to, or greater than $r$. For example, for $x^2+y^2=100$ and the point $(6, -8)$: $6^2+(-8)^2=100$, so it lies exactly on the circle.
If the endpoints of a diameter are $A$ and $B$, the center is their midpoint and the radius is half of $AB$. If the circle is tangent to both axes in the first quadrant, the center is $(r, r)$.
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…
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.
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…
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…
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.
The General Equation
If you expand the squared binomials in the standard equation, you get a form with no parentheses.
To go back: the center is $C\left(-\dfrac{D}{2}, -\dfrac{E}{2}\right)$ and $r^2=\left(\dfrac{D}{2}\right)^2+\left(\dfrac{E}{2}\right)^2-F$.
Another way to go from the general form to the standard form is completing the square: you group the terms in $x$ and in $y$ and complete a perfect square trinomial in each group.
Find the center and radius of the circle describing the outer edge of a gear in a computer-aided design blueprint, whose general equation is $x^2+y^2-6x+8y=0$.
Find the center 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…
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 center and the value of its radius squared.
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…
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 $(1, 1)$.
A Circle Through Three Points
If you know three points, you substitute each one into $x^2+y^2+Dx+Ey+F=0$ and get a system of three equations in $D$, $E$, $F$.
You can also use the fact that the center lies on the perpendicular bisectors of the sides (exercises with a condition on the center, such as lying on a line, are solved the same way: center $(h, k)$ on the line and equal distances to the points).
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…
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 $(0, 3)$.
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…
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 $(0, 2)$.
Find the equation of the circumscribed circle of the triangular park bounded by the avenues whose equations are $x=0$, $y=0$ and $3x+4y-12=0$.
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 Line and a Circle
A line can be a secant (two points in common), a tangent (only one, and it is perpendicular to the radius at that point), or exterior (none).
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…
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…
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…
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…
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 $y=6$.
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…
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…
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…
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).
Two Circles
With distance between centers $d$ and radii $r_1 > r_2$:
| Condition | Position |
|---|---|
| $d > r_1+r_2$ | externally separate |
| $d = r_1+r_2$ | externally tangent |
| $r1-r2 < d < r_1+r_2$ | intersecting (two points) |
| $d = r_1-r_2$ | internally tangent |
| $d < r_1-r_2$ | one inside the other |
If they intersect, subtracting their general equations eliminates $x^2$ and $y^2$ and gives a line: the radical axis, which contains the intersection points.
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 $C_2: (x-4)^2+y^2=4$.
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 $C_2: (x-4)^2+y^2=4$.
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 $x^2+y^2+6x-2y+6=0$.
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 $x^2+y^2+6x-2y+6=0$.
The GPS system of a cell phone receives signals from two telecommunications towers. The first tower is at the origin of coordinates and estimates that the cell phone is…
The GPS system of a cell phone receives signals from two telecommunications towers. The first tower is at the origin of coordinates and estimates that the cell phone is $5$ km away. The second tower is at the point $(5, 0)$ and estimates that the cell phone is $4$ km away. Find the exact coordinates of the two possible points where the cell phone could be located by calculating the intersection of both circles.
Loci That Are Circles
Sometimes a circle shows up as the set of points that satisfy a condition on distances. The method: call the point $P(x, y)$, write the condition with the distance formula, eliminate the roots by squaring, and recognize the form $x^2+y^2+Dx+Ey+F=0$.
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…
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 center and its radius.
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…
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.
Back to the beginning: your cell phone is inside the antenna's coverage area if $d(P, C)