Conic sections and loci
A telecommunications aerial, the orbit of a planet, the cable of a suspension bridge and the reflector of a lighthouse have nothing to do with one another. And yet the curves they describe are only four, and all four come from cutting a cone with a plane. You are going to meet them as loci: the set of points that satisfy a condition on distances. You already know the circle from 3rd ESO; now come the ellipse, the hyperbola and the parabola.
Cutting a cone: where conics 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.
Revision: the locus as a tool
Revise the circle as a locus: drag P until $OP$ is always the same length.
The four conics in this topic are slightly more elaborate loci: instead of a single fixed distance, they impose a relationship between two distances (to one or 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 formulae, and simplify until the roots 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 centre and radius.
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.
Conditions with squared distances
Some conditions come without roots already: they ask for the sum or the difference of the squares of two distances. They are even 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 $20$.
Eccentricity: one recipe for all four curves
This number $e$ —the eccentricity— will reappear in each of the next three sections: it measures how "flattened" the curve is compared with its simplest form, the 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.
Back to the beginning: each application uses a different curve: the orbits of the planets are ellipses; the reflectors of lighthouses and the cables of bridges take parabolic shapes; and the synchronisation of signals between aerials relies on the hyperbola and the parabola.