Ellipse
Stick two thumbtacks into a board, tie the ends of a string to each one, and trace with a pencil while keeping the string taut the whole time: you are drawing an ellipse. It is the curve of the orbits of planets and satellites (with the Sun at a focus, not at the center) and of rooms with a "whispering effect," where two people far apart hear each other as if they were side by side. Why does that whisper work? We will explain it at the end.
The ellipse as a locus
To recall what "locus" means, move the point in this review with the circle:
Given the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$, determine the coordinates of its vertices, those of its foci, and the value of its eccentricity.
Given the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$, determine the coordinates of its vertices, those of its foci, and the value of its eccentricity.
Elements and standard equation
Write the standard equation of an ellipse centred at the origin of coordinates, knowing that one of its foci is the point $F(4, 0)$ and that the length of its major axis…
Write the standard equation of an ellipse centred at the origin of coordinates, knowing that one of its foci is the point $F(4, 0)$ and that the length of its major axis is $10$.
An ellipse has its center at the point $C(-2, 3)$, its major axis is parallel to the x-axis and measures $12$ units, and its minor axis measures $8$ units. Write its…
An ellipse has its center at the point $C(-2, 3)$, its major axis is parallel to the x-axis and measures $12$ units, and its minor axis measures $8$ units. Write its equation in canonical form.
Find the equation of the locus of points in the plane $P(x, y)$ whose sum of distances to the points $F_1(-3, 0)$ and $F_2(3, 0)$ is always equal to $10$. Simplify the…
Find the equation of the locus of points in the plane $P(x, y)$ whose sum of distances to the points $F_1(-3, 0)$ and $F_2(3, 0)$ is always equal to $10$. Simplify the expression to obtain the standard equation of an ellipse.
Eccentricity: how flattened the ellipse is
Calculate the eccentricity of the ellipse whose equation is $16x^2 + 25y^2 = 400$.
Calculate the eccentricity of the ellipse whose equation is $16x^2 + 25y^2 = 400$.
Find the equation of an ellipse centred at the origin, whose foci lie on the y-axis, knowing that the focal distance is $8$ units and the eccentricity is $0.8$.
Find the equation of an ellipse centred at the origin, whose foci lie on the y-axis, knowing that the focal distance is $8$ units and the eccentricity is $0.8$.
The orbit of Mercury around the Sun is an ellipse with the Sun at one of its foci. The minimum distance from Mercury to the Sun (perihelion) is $46$ million kilometers,…
The orbit of Mercury around the Sun is an ellipse with the Sun at one of its foci. The minimum distance from Mercury to the Sun (perihelion) is $46$ million kilometers, and the maximum distance (aphelion) is $70$ million kilometers. Calculate the eccentricity of Mercury's orbit and the length of its semi-minor axis in million kilometers.
Points of the ellipse: interior, exterior, and problems with the equation
To find out whether a point $(x_1,y_1)$ is on the ellipse, inside, or outside, substitute its coordinates into the left side of $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}$:
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If the result is $1$, the point is on the ellipse.
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If it is less than $1$, it is interior.
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If it is greater than $1$, it is exterior.
Why? The farther from the center, the larger $x^2$ and $y^2$ are, and the larger the value of the expression.
This same substitution is used to find an unknown coordinate of a point on the ellipse, such as the height of an arch at a certain distance from the center.
Check whether the points $P(2, -3)$, $Q(5, 0)$ and $R(4, 2)$ lie inside, outside, or on the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$.
Check whether the points $P(2, -3)$, $Q(5, 0)$ and $R(4, 2)$ lie inside, outside, or on the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$.
A stone bridge over a river has the shape of a semi-elliptical arch. The width of the river (major axis of the ellipse) is $20\text{ m}$ and the maximum height of the…
A stone bridge over a river has the shape of a semi-elliptical arch. The width of the river (major axis of the ellipse) is $20\text{ m}$ and the maximum height of the arch at the center of the river is $6\text{ m}$. Calculate the height of the arch at a distance of $4\text{ m}$ from the center of the river.
In a clockwork mechanism, a gear has an elliptical shape. It is known that its boundary equation is given by $x^2 + 4y^2 = 16$. A small pin is to be placed at a point on…
In a clockwork mechanism, a gear has an elliptical shape. It is known that its boundary equation is given by $x^2 + 4y^2 = 16$. A small pin is to be placed at a point on the ellipse whose x-coordinate is twice its y-coordinate. Calculate the coordinates of this point (assuming both are positive) and the exact distance from this point to the center of the gear.
Ellipses with foci and the center: semi-axes, foci, and distances
Many problems give data other than $a$ and $b$, and you have to reconstruct them:
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Total length of the major axis $=2a$; maximum width $=2b$.
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Minimum and maximum distance from a focus: $a-c$ and $a+c$. That is why $a=\dfrac{\text{max}+\text{min}}{2}$ and $c=\dfrac{\text{max}-\text{min}}{2}$.
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Known eccentricity: $c=ea$ and $b^2=a^2-c^2$.
Back to the beginning. In an elliptical room, sound emitted at one focus bounces off the walls and reaches the other focus, because all the paths focus → wall → focus have the same length, $2a$. That is why a whisper can be heard from far away.
The eccentricity of an ellipse is $\frac{1}{3}$ and its semi-minor axis measures $4\sqrt{2}$. Calculate the measure of its semi-major axis.
The eccentricity of an ellipse is $\frac{1}{3}$ and its semi-minor axis measures $4\sqrt{2}$. Calculate the measure of its semi-major axis.
In a whispering gallery, which has the shape of an ellipse, two people located at the foci can hear each other perfectly even if they speak in a whisper. If the gallery…
In a whispering gallery, which has the shape of an ellipse, two people located at the foci can hear each other perfectly even if they speak in a whisper. If the gallery has a total length of $34\text{ m}$ and a maximum width of $16\text{ m}$, at what distance from the center of the room should each person stand to take advantage of this acoustic phenomenon?
A main square is shaped like an ellipse. A straight path is to be paved connecting the leftmost focus, $F_1$, with a point $P$ on the ellipse located in the second…
A main square is shaped like an ellipse. A straight path is to be paved connecting the leftmost focus, $F_1$, with a point $P$ on the ellipse located in the second quadrant. It is known that the major axis of the square measures $100\text{ m}$ and the minor axis measures $60\text{ m}$. If the angle formed by the segment $F_1P$ with the positive x-axis is $60^\circ$, calculate the exact length of the paved path.
General equation and translated ellipses
If the center is $(h,k)$, the equation is translated by replacing $x$ with $x-h$ and $y$ with $y-k$:
$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$The center is $(h,k)$; the foci are at distance $c$ from the center along the major axis (parallel to the $x$-axis if $a^2$ is under $x$, or to the $y$-axis in the other case).
Expanding the squares gives the general form $Ax^2+Cy^2+Dx+Ey+F=0$, with $A$ and $C$ of the same sign (if they are equal, it is a circle, which is a particular case of an ellipse).
From the equation of the ellipse $\frac{(x-1)^2}{16} + \frac{(y+4)^2}{9} = 1$, state the coordinates of its center and the lengths of its semi-axes.
From the equation of the ellipse $\frac{(x-1)^2}{16} + \frac{(y+4)^2}{9} = 1$, state the coordinates of its center and the lengths of its semi-axes.
Given the general equation $9x^2 + 4y^2 - 36x + 24y + 36 = 0$, prove that it corresponds to an ellipse. To do so, express it in canonical form and determine the…
Given the general equation $9x^2 + 4y^2 - 36x + 24y + 36 = 0$, prove that it corresponds to an ellipse. To do so, express it in canonical form and determine the coordinates of its center, its vertices, and its foci.
The ellipse as a locus: finding its equation
To derive the equation from a condition on distances, write the condition with $P(x,y)$, isolate one radical, and square twice.
There are also loci that turn out to be ellipses without being stated as such: if the distance to a focus is a fraction $e<1$ of the distance to a line (the directrix), the locus is an ellipse of eccentricity $e$. And if each point of a circle of radius $r$ has its ordinate multiplied by $k$ ($0 Another source of ellipses is tangent circles. If a circle with center $C$ and radius $r$ is internally tangent to another with center $O$ and radius $R$ (the small one inside, touching the large one at one point), the distance between the centers is the difference of the radii: Why? The point of tangency lies on the line joining the centers. From $O$ to that point there is $R$, and from that point to $C$ there is $r$, so $O$ and $C$ are separated by $R-r$. If in addition the small circle passes through a fixed point $A$, its radius is $r=d(C,A)$. Then $d(O,C)=R-d(C,A)$, that is, $d(C,O)+d(C,A)=R$: the sum of the distances from $C$ to two fixed points is constant, and that is an ellipse with foci $O$ and $A$. Determine the locus of points in the plane whose distance to the point $F(4, 0)$ is equal to half their distance to the line with equation $x - 16 = 0$. Identify the resulting conic section and calculate its center and the lengths of its semi-axes. Consider the circle with equation $x^2 + y^2 = 100$. With each point $A(x, y)$ on the circle, we associate a point $P(x, y')$ that has the same x-coordinate, but its y-coordinate is $60\%$ of the y-coordinate of $A$. Prove that the locus of the points $P$ is an ellipse and determine its equation, its center and its foci. Find the locus of the centers of the circles that are internally tangent to the circle with equation $x^2 + y^2 = 100$ and pass through the fixed point $A(6, 0)$. Prove that this locus is an ellipse, find its canonical equation and determine the coordinates of its foci.Internally tangent circles
Determine the locus of points in the plane whose distance to the point $F(4, 0)$ is equal to half their distance to the line with equation $x - 16 = 0$. Identify the…
Consider the circle with equation $x^2 + y^2 = 100$. With each point $A(x, y)$ on the circle, we associate a point $P(x, y')$ that has the same x-coordinate, but its…
Find the locus of the centers of the circles that are internally tangent to the circle with equation $x^2 + y^2 = 100$ and pass through the fixed point $A(6, 0)$. Prove…
Ellipse and line: relative position and tangents
To see how a line and an ellipse intersect, solve for one variable from the line, substitute it into the ellipse's equation, and solve the quadratic. Its discriminant $\Delta$ decides:
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$\Delta>0$: secant (two intersection points).
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$\Delta=0$: tangent (a single point).
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$\Delta<0$: exterior (they do not intersect).
For tangents we impose $\Delta=0$. If the tangent has slope $m$ and the line is $y=mx+n$, on $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ the condition is $n^2=a^2m^2+b^2$.
Study the relative position of the line with equation $2x + y - 5 = 0$ and the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. If they intersect, find the coordinates of…
Study the relative position of the line with equation $2x + y - 5 = 0$ and the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. If they intersect, find the coordinates of the points of intersection.
Find the equations of the tangent lines to the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$ that pass through the exterior point $P(0, 5)$.
Find the equations of the tangent lines to the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$ that pass through the exterior point $P(0, 5)$.
Determine the equations of the tangent lines to the ellipse $x^2 + 2y^2 = 18$ that are parallel to the line with equation $x - y + 4 = 0$.
Determine the equations of the tangent lines to the ellipse $x^2 + 2y^2 = 18$ that are parallel to the line with equation $x - y + 4 = 0$.
Calculate the values of the parameter $k$ such that the line $x + 2y + k = 0$ is tangent to the ellipse with equation $x^2 + 4y^2 = 20$.
Calculate the values of the parameter $k$ such that the line $x + 2y + k = 0$ is tangent to the ellipse with equation $x^2 + 4y^2 = 20$.
Determining the ellipse from data and optimization problems
Sometimes the ellipse is not given: you have to find it. If its axes are the coordinate axes, the equation is $\dfrac{x^2}{p}+\dfrac{y^2}{q}=1$ and each known point gives a linear equation in $\dfrac1p$ and $\dfrac1q$.
For a rectangle with sides parallel to the axes inscribed in $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, if one vertex is $(x,y)$ its dimensions are $2x$ and $2y$ and the area is $4xy$.
A rectangular window is to be designed, inscribed in an elliptical opening with equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. The sides of the rectangle must be…
A rectangular window is to be designed, inscribed in an elliptical opening with equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. The sides of the rectangle must be parallel to the axes of the ellipse. If the rectangle is required to have a total area of $24\text{ m}^2$, what should its exact dimensions be?
An ellipse is centred at the origin and its axes of symmetry coincide with the Cartesian coordinate axes. Given that the ellipse passes through the points…
An ellipse is centred at the origin and its axes of symmetry coincide with the Cartesian coordinate axes. Given that the ellipse passes through the points $A(2, \sqrt{3})$ and $B(\sqrt{2}, \frac{\sqrt{15}}{2})$, find the equation of the ellipse and calculate the exact distance between its foci.
Investigate with dynamic geometry
Before constructing, predict: if the foci move apart while the sum of distances stays constant, will the ellipse be more circular or more elongated?
In GeoGebra (or another dynamic geometry program), create a slider $c$ between $0$ and $4.5$, fix $a=5$, and place $F_1=(-c,0)$ and $F_2=(c,0)$. Graph $x^2/25+y^2/(25-c^2)=1$. Place a point $P$ on the curve and measure its distances to the foci. As you move $P$, record the sum; as you change $c$, record the vertical semi-axis.
Both ellipses have $a=5$; the narrower one has the foci farther apart:
Present a screenshot or a table with your conclusion. Do not use $c=5$: it would make a denominator zero and the ellipse would degenerate into a segment. A real ellipse requires $2a>2c$.