Equations of Circles
Your phone drops its signal once you are more than 6 miles from a cell tower. On a map where the tower sits at the origin and each unit is 1 mile, your cabin is 4 miles east and 5 miles north of the tower. Do you get service? Circles answer this kind of question: once a circle has an equation, you can test points, lines, and other circles with algebra.
Standard Form
A circle is the set of all points in a plane that are the same distance from one fixed point. That fixed point is the center $C(h, k)$, and the distance is the radius $r$.
Why does it work? The distance from $P(x, y)$ to $C(h, k)$ is $\sqrt{(x-h)^2+(y-k)^2}$. Setting that equal to $r$ and squaring removes the square root.
Is a point inside, on, or outside?
Compare the point's squared distance from the center with $r^2$. Plug $(x, y)$ into $(x-h)^2+(y-k)^2$:
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result $< r^2$: the point is inside the circle,
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result $= r^2$: the point is on the circle,
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result $> r^2$: the point is outside the circle.
Back to the start: the cabin is $\sqrt{41}\approx 6.4$ miles from the tower, just past the 6-mile limit.
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…
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…
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.…
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.
Writing the Equation from Given Information
Whatever the problem gives you, your goal is always the same: find the center $(h, k)$ and the radius $r$. Here are the common situations.
| You are given | Center | Radius |
|---|---|---|
| Center and radius | given | given |
| Center and a point on the circle | given | distance from center to the point |
| Endpoints of a diameter | midpoint of the diameter | distance from center to an endpoint (or half the diameter) |
| Circle tangent to the axes | distance from each axis equals $r$ | given |
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…
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 $(2, 0)$.
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…
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 $(7, -6)$.
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…
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.
General Form and Completing the Square
If you multiply out the squares in standard form, the parentheses disappear:
Why? Expanding gives $x^2-2hx+h^2+y^2-2ky+k^2-r^2=0$. Matching terms with $x^2+y^2+Dx+Ey+F=0$ gives the values above.
To go the other way, complete the square in $x$ and in $y$. Add $\left(\tfrac{\text{coefficient}}{2}\right)^2$ to both sides for each variable.
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 $x^2+y^2-6x+8y=0$.
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 $x^2+y^2-6x+8y=0$.
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…
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.
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 $x^2+y^2-10x+4y+13=0$…
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 $(1, 1)$.
Circles Through Points or with a Center Condition
Sometimes you are not given the center. Two strategies work.
Strategy 1: three points. Plug each point into $x^2+y^2+Dx+Ey+F=0$. You get three linear equations in $D$, $E$, $F$. Solve them, then complete the square. This is also how you find the circumscribed circle of a triangle: the circle through its three vertices. If the vertices are not given, find them first by intersecting the sides.
Strategy 2: center on a line. Call the center $(h, k)$ and use the line to write $k$ in terms of $h$. The center is equally far from any two points of the circle, so set those two squared distances equal and solve.
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…
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 $(0, 3)$.
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…
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 $(0, 2)$.
Find the equation of the circumscribed circle to the triangular park bounded by the avenues whose equations are $x=0$, $y=0$ and $3x+4y-12=0$.
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 Line and a Circle
A line and a circle can have two common points (secant), exactly one (tangent), or none (exterior).
There are two ways to decide, and both are worth knowing.
Method 1: substitute. Solve the line for $y$ (or $x$) and plug it into the circle. You get a quadratic. Its discriminant tells you the number of solutions: $\Delta>0$ means secant, $\Delta=0$ means tangent, $\Delta<0$ means exterior. Its solutions give the intersection points.
Method 2: compare distances. Find the distance $d$ from the center to the line and compare it with $r$.
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…
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 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…
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 $y=6$.
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…
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 $4x-3y+2=0$.
Tangent Line at a Point
A tangent line touches the circle at one point and is perpendicular to the radius drawn to that point. So its slope is the negative reciprocal of the radius's slope.
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…
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 $(3, 4)$.
Tangent Lines from Other Conditions
Often you are not given the point of tangency. Instead you get a point outside the circle, a direction (parallel to a given line), or a line with an unknown parameter. In all three cases, use the same idea: the distance from the center to the tangent line equals $r$.
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From an external point: write a line through the point with unknown slope $m$, then set distance $=r$. There are two tangent lines. If the point is on the circle there is one, and if it is inside there are none.
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Parallel to a given line: a line parallel to $Ax+By+C_0=0$ has the form $Ax+By+c=0$ with the same $A$ and $B$. Set distance $=r$ and solve for $c$.
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A family with parameter $k$: set distance $=r$ (or set the discriminant of the substituted quadratic equal to $0$) and solve for $k$.
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…
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).
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…
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.
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…
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.
Two Circles
Two circles are related by two numbers: the distance $d$ between their centers, and their radii $r_1$ and $r_2$.
To find centers and radii, put each equation in standard form first (complete the square if needed).
When two circles intersect, subtract their equations (both written in general form). The $x^2$ and $y^2$ terms cancel and you get a line through the intersection points. Some problems call this line the radical axis. Substitute it into either circle to find the points.
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 $C_2: (x-4)^2+y^2=4$.
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 $C_2: (x-4)^2+y^2=4$.
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 $x^2+y^2+6x-2y+6=0$.
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 $x^2+y^2+6x-2y+6=0$.
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 $5$…
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.
Circles as Loci
A locus is the set of all points that satisfy a geometric condition. Some loci turn out to be circles. To prove it and find the equation:
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Let $P(x, y)$ be a general point.
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Write the condition with the distance formula.
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Square both sides to remove square roots, expand, and simplify.
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Complete the square and recognize $(x-h)^2+(y-k)^2=r^2$ with $r^2>0$.
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…
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.
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…
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.