Conic Sections and Loci
A satellite dish, the orbit of a planet, the cable of a suspension bridge, and a lighthouse reflector seem to have nothing in common. Yet the curves they follow are only four, and all four come from cutting a cone with a plane. On this page you will meet them as loci: the set of points that satisfy a condition on distances. You already know the circle. Now come the ellipse, the hyperbola, and the parabola.
Cutting a cone: where conic sections come from
A cone of revolution is cut by a plane that does not pass through its vertex. If the plane is parallel to a single generator of the cone, what conic section is obtained?…
A cone of revolution is cut by a plane that does not pass through its vertex. If the plane is parallel to a single generator of the cone, what conic section is obtained? Justify your answer based on the geometric definition of conic sections.
The locus as a tool
Drag the point $P$ and observe what the distance $OP$ does in the circle:
Now the perpendicular bisector: the points that are equidistant from $A$ and $B$.
Conic sections are somewhat more elaborate loci. Instead of a single fixed distance, they impose a relationship between two distances, to two fixed points or to a point and a line.
Find the equation of the locus of points in the plane whose distance from the origin of coordinates is always equal to $5$.
Find the equation of the locus of points in the plane whose distance from the origin of coordinates is always equal to $5$.
Find the equation of the locus of points in the plane that are equidistant from points $A(2, -3)$ and $B(-4, 5)$. What geometric shape does it represent?
Find the equation of the locus of points in the plane that are equidistant from points $A(2, -3)$ and $B(-4, 5)$. What geometric shape does it represent?
Building the equation step by step
The method is always the same. You call the general point $P(x, y)$, write the condition using the distance formula between two points, and simplify until the radicals disappear.
In a telecommunications project, two transmitting antennas are located at points $A(-3.0)$ and $B(3,0)$ on a Cartesian plane (in km). A receiver $R(x,y)$ moves such that…
In a telecommunications project, two transmitting antennas are located at points $A(-3.0)$ and $B(3,0)$ on a Cartesian plane (in km). A receiver $R(x,y)$ moves such that its distance to antenna $A$ is always twice its distance to antenna $B$. Find the equation of the locus described by the receiver and identify which conic section it is, stating its center and radius.
Conditions with squared distances
Some conditions already come without radicals: they ask for the sum or the difference of the squares of two distances. They are quicker to work with.
Find the locus of points $P(x,y)$ whose difference of squared distances to points $A(1,0)$ and $B(-1.0)$ is equal to $8$, that is, $d(P,A)^2 - d(P,B)^2 = 8$.
Find the locus of points $P(x,y)$ whose difference of squared distances to points $A(1,0)$ and $B(-1.0)$ is equal to $8$, that is, $d(P,A)^2 - d(P,B)^2 = 8$.
Find the equation of the locus of points $P(x,y)$ such that the sum of the squares of their distances to points $A(2,0)$ and $B(-2.0)$ is equal to $20$.
Find the equation of the locus of points $P(x,y)$ such that the sum of the squares of their distances to points $A(2,0)$ and $B(-2.0)$ is equal to $20$.
Three seismic sensors are located at coordinates $A(0,0)$, $B(4,0)$ and $C(0,4)$. A calibration drone flies in the same plane such that the sum of the squares of its…
Three seismic sensors are located at coordinates $A(0,0)$, $B(4,0)$ and $C(0,4)$. A calibration drone flies in the same plane such that the sum of the squares of its distances to the three sensors remains constant and equal to $50$. Find the equation of the locus of the drone's path and describe the resulting shape.
A sailboat navigates such that the difference of the squares of its distances to two buoys located at $A(-3, -1)$ and $B(5, 5)$ is always constant and equal to $32$.…
A sailboat navigates such that the difference of the squares of its distances to two buoys located at $A(-3, -1)$ and $B(5, 5)$ is always constant and equal to $32$. That is, $d(P,A)^2 - d(P,B)^2 = 32$. Find the equation of the locus of the sailboat's trajectory and prove analytically that this trajectory is a line perpendicular to the segment joining the two buoys.
Distance to a line: bisectors and parallels
When the condition involves lines, we use the distance from a point $P(x_0, y_0)$ to the line $Ax+By+C=0$:
$$d(P,r)=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$$This formula is justified using the perpendicular projection of $P$ onto the line. Here it is enough to use it.
See how the angle bisector forms as a locus:
Determine the locus of points equidistant from the parallel lines $r: x + 2y - 4 = 0$ and $s: x + 2y + 6 = 0$.
Determine the locus of points equidistant from the parallel lines $r: x + 2y - 4 = 0$ and $s: x + 2y + 6 = 0$.
Determine the equations of the loci of points that are equidistant from the intersecting lines $r: 3x - 4y + 5 = 0$ and $s: 4x + 3y - 1 = 0$.
Determine the equations of the loci of points that are equidistant from the intersecting lines $r: 3x - 4y + 5 = 0$ and $s: 4x + 3y - 1 = 0$.
Find the equations of the locus of points in the plane that are at a constant distance of $3$ units from the line $4x - 3y + 2 = 0$.
Find the equations of the locus of points in the plane that are at a constant distance of $3$ units from the line $4x - 3y + 2 = 0$.
On a city map, two straight avenues follow the equations $r: 2x + y - 4 = 0$ and $s: x - 2y + 3 = 0$. It is desired to build a town square at a point equidistant from…
On a city map, two straight avenues follow the equations $r: 2x + y - 4 = 0$ and $s: x - 2y + 3 = 0$. It is desired to build a town square at a point equidistant from both avenues and which, due to planning restrictions, is situated on the pedestrian street $x + y = 5$. Calculate the coordinates of the possible locations for the square.
Sum or difference of distances: ellipse and hyperbola
The ellipse can be drawn: drive two stakes (the foci), tie a string around them, and trace the outline with the string pulled taut. The length of the string is the constant sum.
To design an elliptical flowerbed in a park, a gardener drives two stakes into the ground at points $(-3.0)$ and $(3,0)$ (measurements in meters). Then, they tie the…
To design an elliptical flowerbed in a park, a gardener drives two stakes into the ground at points $(-3.0)$ and $(3,0)$ (measurements in meters). Then, they tie the ends of a rope of length $10\text{ m}$ to both stakes and, keeping the rope taut with a stylus, trace the outline on the ground. Find the equation of the locus of the edge of the flowerbed.
In a hyperbolic navigation system, a ship receives simultaneous radio signals from two coastal stations located at $A(-5.0)$ and $B(5,0)$ (in tens of km). The delay in…
In a hyperbolic navigation system, a ship receives simultaneous radio signals from two coastal stations located at $A(-5.0)$ and $B(5,0)$ (in tens of km). The delay in receiving the signal indicates that the ship is always $60\text{ km}$ closer to station $B$ than to station $A$. Obtain the equation of the locus on which the ship is sailing.
A laser beam leaves focus $F_1(c,0)$ of a reflective ellipse, bounces off a point $P(x,y)$ on its surface and arrives directly at the other focus $F_2(-c,0)$. If the…
A laser beam leaves focus $F_1(c,0)$ of a reflective ellipse, bounces off a point $P(x,y)$ on its surface and arrives directly at the other focus $F_2(-c,0)$. If the total distance travelled by the light beam is $10$ units and the focal distance is $6$ units, deduce the standard equation of the locus of the ellipse using its definition based on distances.
Eccentricity: one recipe for all four curves
This number $e$ shows up again on the pages about the ellipse, the parabola, and the hyperbola. It measures how "open" the curve is compared to a circle.
The reflector of a lighthouse has a profile designed such that the distance from any point on its surface to the light-emitting focus, situated at $F(0,2)$, is exactly…
The reflector of a lighthouse has a profile designed such that the distance from any point on its surface to the light-emitting focus, situated at $F(0,2)$, is exactly equal to its distance from a horizontal reference line with equation $y = -2$. Determine the equation of the locus of the reflector's profile.
The end $P(x,y)$ of an industrial robotic arm is programmed to move such that its distance to the safety line $x = -4$ is always twice its distance to the anchor point…
The end $P(x,y)$ of an industrial robotic arm is programmed to move such that its distance to the safety line $x = -4$ is always twice its distance to the anchor point $F(-1.0)$. Determine the equation of the path described by the end of the arm and identify what type of conic section it represents.
A moving object travels in the Cartesian plane such that its distance from the vertical line $x = k$ is equal to its distance from the fixed point $F(2,0)$. Determine…
A moving object travels in the Cartesian plane such that its distance from the vertical line $x = k$ is equal to its distance from the fixed point $F(2,0)$. Determine the equation of the locus in terms of the parameter $k$. For what exact value of $k$ does the locus degenerate into a line instead of forming a parabola?
Prove analytically that the locus of points $P(x,y)$ whose distance from the origin $O(0,0)$ is equal to $e$ times their distance from the line $x = d$ (with $e > 0$ and…
Prove analytically that the locus of points $P(x,y)$ whose distance from the origin $O(0,0)$ is equal to $e$ times their distance from the line $x = d$ (with $e > 0$ and $d > 0$) represents an ellipse if $e < 1$, a parabola if $e = 1$ and a hyperbola if $e > 1$. Write the general equation obtained before classifying it.
Other conditions: slopes, angles, and tangents
The method does not change: general points, the condition written in $x$ and $y$, and simplifying. These are other conditions that appear in the exercises.
During an archaeological excavation, a fragment of the rim of an ancient circular plate is discovered. To reconstruct its original size, the fragment is digitized and…
During an archaeological excavation, a fragment of the rim of an ancient circular plate is discovered. To reconstruct its original size, the fragment is digitized and three points on its contour are taken: $A(-2, 1)$, $B(2, 5)$ and $C(4, 1)$. Using the concept of locus (intersection of perpendicular bisectors), calculate the coordinates of the original center of the plate and its radius.
A security camera located at a moving point $P(x,y)$ must monitor a linear corridor defined by the segment $AB$, with endpoints at $A(-4.0)$ and $B(4,0)$. For the…
A security camera located at a moving point $P(x,y)$ must monitor a linear corridor defined by the segment $AB$, with endpoints at $A(-4.0)$ and $B(4,0)$. For the camera's sensors to function optimally, the viewing angle $\angle APB$ must be exactly $90^\circ$. Find analytically the locus of the possible positions of the camera, excluding points $A$ and $B$. Describe the resulting shape.
In the design of a fluid system, a cylindrical pipe must be installed whose cross-section can vary in size, but must always be tangent to the x-axis and must pass…
In the design of a fluid system, a cylindrical pipe must be installed whose cross-section can vary in size, but must always be tangent to the x-axis and must pass through a fixed sensor located at $P(0,2)$. Find the equation of the locus described by the possible centers $C(x,y)$ of this cross-section and identify the conic section.
In an acoustic study, two sound sources emit waves along perpendicular axes that coincide with the coordinate axes. The locus of points where the product of the…
In an acoustic study, two sound sources emit waves along perpendicular axes that coincide with the coordinate axes. The locus of points where the product of the distances to both axes is constant and equal to $16$ defines isophonic curves. Find the equation of the locus and prove that it consists of two rectangular hyperbolas, calculating the exact distance between the vertices of each of them.
In an articulated mechanism, two rods rotate around fixed pivots $A(-a,0)$ and $B(a,0)$. The mechanism is designed such that the product of the gradients of both rods at…
In an articulated mechanism, two rods rotate around fixed pivots $A(-a,0)$ and $B(a,0)$. The mechanism is designed such that the product of the gradients of both rods at their point of intersection $P(x,y)$ is always a negative constant $-k^2$ (with $k > 0$). Prove that the locus of point $P$ is an ellipse and determine the length of its semi-axes in terms of $a$ and $k$.
Back to the beginning: each application uses a different curve. The orbits of the planets are ellipses, satellite dishes and lighthouse reflectors have a parabolic profile, and the hyperbolic navigation of coastal stations uses hyperbolas. All four curves are loci of a condition on distances, and all of them come from cutting a cone.