A lighthouse reflector, a satellite dish, the main cable of the Golden Gate Bridge and the orbit of a planet have nothing in common. Yet the curves behind them come from just four families, and all four appear when you slice a cone with a plane. In this section you will meet them as loci: sets of points that satisfy a condition about distances. You already know the circle. Now come the parabola, the ellipse and the hyperbola, and one number, the eccentricity, will tell them apart.
Slicing a Cone
A straight line on the cone's surface through the vertex is called a side of the cone (some problems call it a generatrix). If the plane is parallel to exactly one side, you get a parabola. If it is tilted but cuts every side of one nappe (half-cone), you get an ellipse. If it is perpendicular to the axis, you get a circle.
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A cone of revolution is intersected by a plane that does not pass through its vertex. If the plane is parallel to a single generatrix of the cone, what conic section is…
A cone of revolution is intersected by a plane that does not pass through its vertex. If the plane is parallel to a single generatrix of the cone, what conic section is obtained? Justify your answer based on the geometric definition of conic sections.
Drag the point and watch which distances stay equal.
The four conics of this chapter are just more interesting loci. Instead of one fixed distance, they tie together two distances: to two points, or to a point and a line.
Equidistance from three points works the same way. Intersect two perpendicular bisectors and you get the one point that is equally far from all three, the circumcenter. That is how you find the center of a circle through three points.
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Find the equation of the locus of points in the plane whose distance to the origin of coordinates is always equal to $5$.
Find the equation of the locus of points in the plane whose distance to the origin of coordinates is always equal to $5$.
Find the equation of the locus of points in the plane that are equidistant from the points $A(2, -3)$ and $B(-4, 5)$. What geometric figure does it represent?
Find the equation of the locus of points in the plane that are equidistant from the points $A(2, -3)$ and $B(-4, 5)$. What geometric figure does it represent?
During an archaeological excavation, a fragment of the edge 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 edge of an ancient circular plate is discovered. To reconstruct its original size, the fragment is digitized and three points on its outline 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 plate's original center and its radius.
The method never changes: call the general point $P(x, y)$, write the condition with the distance formula, and simplify until the radicals are gone.
That example was not a coincidence. Whenever a point keeps a constant ratio of distances to two fixed points, the locus is a circle, unless the ratio is $1$.
The coordinate method is the one you just used: put $P(x, y)$, replace each distance with the distance formula, square both sides (the $k$ becomes $k^2$), then collect terms and complete the square in $x$ and in $y$.
Once you have an equation with no radicals and no $xy$ term, you can tell the curve from its squared terms.
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In a telecommunications project, two transmitting antennas are located at the points $A(-3,0)$ and $B(3,0)$ of a Cartesian plane (in miles). A receiver $R(x,y)$ moves in…
In a telecommunications project, two transmitting antennas are located at the points $A(-3,0)$ and $B(3,0)$ of a Cartesian plane (in miles). A receiver $R(x,y)$ moves in such a way 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 it is, indicating its center and its radius.
Some loci involve a line instead of a point. For that you need the distance from a point to a line.
Setting that distance equal to a constant $d$ gives $|Ax + By + C| = d\sqrt{A^2 + B^2}$. The absolute value splits into two equations, so the locus is two parallel lines, one on each side of the original.
A neat shortcut: the points equidistant from two parallel lines form a third parallel line halfway between them.
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Determine the locus of points that are equidistant from the parallel lines $r: x + 2y - 4 = 0$ and $s: x + 2y + 6 = 0$.
Determine the locus of points that are equidistant from the parallel lines $r: x + 2y - 4 = 0$ and $s: x + 2y + 6 = 0$.
Points equidistant from two intersecting lines lie on the angle bisectors. Two crossing lines make two pairs of vertical angles, so there are two bisectors, and they are perpendicular to each other.
Setting the two point-to-line distances equal gives
Remove the absolute values with a $\pm$, and each sign gives one bisector.
If the lines have different values of $\sqrt{A^2 + B^2}$, keep both denominators and multiply through (cross-multiply) before splitting into the two cases.
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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 $s: 4x + 3y - 1 = 0$.
On a city map, two straight avenues follow the equations $r: 2x + y - 4 = 0$ and $s: x - 2y + 3 = 0$. A square is to be built at a point that is equidistant from both…
On a city map, two straight avenues follow the equations $r: 2x + y - 4 = 0$ and $s: x - 2y + 3 = 0$. A square is to be built at a point that is equidistant from both avenues and which, due to urban planning restrictions, is located on the pedestrian street $x + y = 5$. Calculate the coordinates of the possible locations for the square.
Some conditions come without radicals: they ask for the sum or the difference of the squares of two distances. They are quick to work with.
This idea also gives a clean way to handle right angles. If $\angle APB = 90^\circ$, then by the Pythagorean Theorem $d(P,A)^2 + d(P,B)^2 = d(A,B)^2$, which is a sum of squares equal to a constant.
The same trick works with three or more points. For $A$, $B$, $C$ the squared terms add up to $3x^2 + 3y^2 + \dots$. Divide by $3$ first, then complete the square for both $x$ and $y$.
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Find the equation of the locus of points $P(x,y)$ such that the sum of the squares of their distances to the 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 the points $A(2,0)$ and $B(-2,0)$ is equal to $20$.
Three seismic sensors are located at the coordinates $A(0,0)$, $B(4,0)$ and $C(0,4)$. A calibration drone flies in the same plane so that the sum of the squares of its…
Three seismic sensors are located at the coordinates $A(0,0)$, $B(4,0)$ and $C(0,4)$. A calibration drone flies in the same plane so 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 trajectory and describe the resulting figure.
A security camera located at a moving point $P(x,y)$ must monitor a linear corridor defined by the line 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 line segment $AB$, with endpoints at $A(-4,0)$ and $B(4,0)$. For the camera's sensors to work optimally, the viewing angle $\angle APB$ must be exactly $90^\circ$. Find analytically the locus of the possible positions of the camera, excluding the points $A$ and $B$. Describe the resulting figure.
The $x^2$ and $y^2$ terms cancel, so the equation is linear.
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Find the locus of points $P(x,y)$ whose difference of squares of distances to the 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 squares of distances to the points $A(1,0)$ and $B(-1,0)$ is equal to $8$, that is, $d(P,A)^2 - d(P,B)^2 = 8$.
A surveying drone flies so that the difference of the squares of its distances to two emitting beacons located at $A(-3, -1)$ and $B(5, 5)$ is always constant and equal…
A surveying drone flies so that the difference of the squares of its distances to two emitting beacons 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 drone's trajectory and prove analytically that this trajectory is a line perpendicular to the line segment joining the two beacons.
Other conditions multiply two quantities. Two common cases are the product of two slopes and the product of the distances to the axes.
Product of slopes. Take $A(-a, 0)$ and $B(a, 0)$ and require $m_{PA} \cdot m_{PB} = m$. Since $m_{PA} = \dfrac{y}{x + a}$ and $m_{PB} = \dfrac{y}{x - a}$, the condition is $\dfrac{y^2}{x^2 - a^2} = m$, so $y^2 = m(x^2 - a^2)$. If $m = -1$ the lines $PA$ and $PB$ are perpendicular and you get the circle $x^2 + y^2 = a^2$ again. If $m < 0$ the signs of $x^2$ and $y^2$ agree, which gives an ellipse. If $m > 0$ they are opposite, which gives a hyperbola.
Product of distances to the axes. A point $P(x, y)$ is $|x|$ from the $y$-axis and $|y|$ from the $x$-axis, so the product of these distances being constant means $|xy| = k$. Take the branch with $xy = k > 0$, in quadrants I and III.
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In an articulated mechanism, two rods rotate around the fixed pivots $A(-a,0)$ and $B(a,0)$. The mechanism is designed so that the product of the slopes of both rods at…
In an articulated mechanism, two rods rotate around the fixed pivots $A(-a,0)$ and $B(a,0)$. The mechanism is designed so that the product of the slopes 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 the point $P$ is an ellipse and determine the length of its semi-axes in terms of $a$ and $k$.
In an acoustic study, two sound sources emit waves on perpendicular axes that coincide with the coordinate axes. The locus of the points where the product of the…
In an acoustic study, two sound sources emit waves on perpendicular axes that coincide with the coordinate axes. The locus of the points where the product of the distances to both axes is constant and equal to $16$ defines equal-loudness contours. 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.
Move the slider and watch how $p$ controls the width: the focus sits $p$ units above the vertex.
The same locus shows up in disguise. A circle tangent to a line $\ell$ and passing through a point $F$ not on $\ell$ has its radius equal both to the distance from its center to $\ell$ and to the distance from its center to $F$. So its center is equidistant from a point and a line, and it moves along a parabola.
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The reflector of a maritime lighthouse has a profile designed so that the distance from any point on its surface to the light-emitting focus, located at $F(0,2)$, is…
The reflector of a maritime lighthouse has a profile designed so that the distance from any point on its surface to the light-emitting focus, located at $F(0,2)$, is exactly equal to its distance to a horizontal reference line with equation $y = -2$. Determine the equation of the locus of the reflector's profile.
A moving particle travels in the Cartesian plane so that its distance to the vertical line $x = k$ is equal to its distance to the fixed point $F(2,0)$. Determine the…
A moving particle travels in the Cartesian plane so that its distance to the vertical line $x = k$ is equal to its distance to 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?
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 pass through a…
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 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.
The number $e$ is the eccentricity. It measures how far a conic is from being a circle. You will meet it again in the next three sections.
Here is why $e$ decides the shape. Put the focus at the origin and the directrix at $x = d$. The condition $\sqrt{x^2 + y^2} = e\,|x - d|$ becomes, after squaring,
$$(1 - e^2)\,x^2 + y^2 + 2e^2d\,x - e^2d^2 = 0$$
The coefficient of $x^2$ is $1 - e^2$, and its sign is what the recognition rule looks at. If $e < 1$, both squared terms are positive. If $e = 1$, the $x^2$ term vanishes. If $e > 1$, the signs are opposite. To classify fully, complete the square in $x$ and look at the signs.
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The end $P(x,y)$ of an industrial robotic arm is programmed to move in such a way that its distance to the safety line $x = -4$ is always twice its distance to the…
The end $P(x,y)$ of an industrial robotic arm is programmed to move in such a way 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 trajectory described by the end of the arm and identify what type of conic it represents.
Prove analytically that the locus of the points $P(x,y)$ whose distance to the origin $O(0,0)$ is equal to $e$ times their distance to the line $x = d$ (with $e > 0$ and…
Prove analytically that the locus of the points $P(x,y)$ whose distance to the origin $O(0,0)$ is equal to $e$ times their distance to 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.
Sum and Difference of Distances: Ellipse and Hyperbola
There is another way to build conics, using two foci and no directrix.
A gardener can draw an ellipse with two stakes and a loop of string: the taut string forces the sum of the distances to the stakes to stay equal to the string's length.
For the hyperbola the very same steps work with $\big|d_1 - d_2\big| = 2a$. The only change is that $c > a$, so we define $b^2 = c^2 - a^2$ and get $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$.
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To design an elliptical flowerbed in a park, a gardener drives two stakes into the ground at the points $(-3,0)$ and $(3,0)$ (measured in feet). Then, they tie the ends…
To design an elliptical flowerbed in a park, a gardener drives two stakes into the ground at the points $(-3,0)$ and $(3,0)$ (measured in feet). Then, they tie the ends of a string of length $10\text{ ft}$ to both stakes and, keeping the string taut with a peg, trace the outline on the earth. 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 miles). The delay…
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 miles). The delay in receiving the signal indicates that the ship is always $60\text{ mi}$ closer to station $B$ than to station $A$. Obtain the equation of the locus on which the ship is sailing.
A laser beam starts from the focus $F_1(c,0)$ of a reflecting ellipse, bounces off a point $P(x,y)$ on its surface and arrives directly at the other focus $F_2(-c,0)$.…
A laser beam starts from the focus $F_1(c,0)$ of a reflecting 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 traveled 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.
Back to the start: each application uses a different curve. Planetary orbits are ellipses. Lighthouse reflectors, satellite dishes and the shape of a suspension cable under a uniform load are parabolas. Navigation systems that compare the arrival times of two signals place you on a branch of a hyperbola. All four are slices of a cone and loci defined by distances.
Summary
Let's practice! 8 medium exercises in this section