Ellipses

The ellipse as a locus

The string stays tight, so the pencil is always at a point $P$ where (distance to thumbtack 1) + (distance to thumbtack 2) equals the string length minus the gap between the tacks. That total never changes. This is the definition.

The constant $2a$ equals the string length minus the gap between the tacks (the distance $2c$ between the foci). It is also the length of the major axis, the longest chord of the ellipse.

Now let's turn the definition into an equation. This is exactly what "find the locus" problems ask you to do.

Standard equation centered at the origin

When the center is the origin and the axes of the ellipse lie along the coordinate axes, the equation is very clean.

Some books call all four endpoints of the axes "vertices." Here the vertices are the endpoints of the major axis, and the co-vertices are the endpoints of the minor axis. Some problems use the phrase focal distance for the distance $2c$ between the two foci.

Eccentricity: how flat is the ellipse?

Two ellipses can both have $a = 5$ and still look very different. The foci can be close together (nearly a circle) or far apart (long and thin). Eccentricity measures this.

Try it yourself. Here $a = 5$ is fixed. Drag $b$ from $5$ down to $0.5$ and watch the shape, then think about where the foci would be.

Ellipses not centered at the origin

If the center is $(h, k)$, replace $x$ with $x - h$ and $y$ with $y - k$. This is the same shift you used for circles.

Problems often hand you a general equation $Ax^2 + By^2 + Cx + Dy + E = 0$. To get the center and axes, complete the square in $x$ and in $y$ separately.

Points and lines versus an ellipse

Is a point inside, on, or outside?

The ellipse is the set where $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. Moving a point outward makes $x$ and $y$ larger in absolute value, so the left side grows. That gives a quick test.

Where does a line cross the ellipse?

Substitute the line into the ellipse equation. You get a quadratic, and its discriminant tells you what happens.

The same idea works when a condition ties $x$ and $y$ together and you need a point on the ellipse. Substitute the condition into the equation, solve, then go back for the other coordinate. For example, on $x^2 + 4y^2 = 20$ with $y = 3x$ you get $x^2 + 36x^2 = 20$, which gives $x^2 = \tfrac{20}{37}$.

Ellipses in the real world

To use an ellipse in a real problem: place the center at the origin, line up the major axis with an axis, find $a$ and $b$, and write the equation. Then answer the question with the equation.

Orbits: closest and farthest

If the Sun is at one focus, the point of the orbit closest to the Sun is a vertex, and so is the farthest point. The distances are:

Adding gives $2a$, and subtracting gives $2c$.

Arches

Whispering galleries

All the paths from one focus to the wall and then to the other focus have the same length, $2a$. Sound that leaves one focus in any direction therefore arrives at the other focus at the same instant, and the waves reinforce each other. That is why a whisper at one focus can be heard at the other.

Distances from a point to the foci

For any point $P$ on the ellipse, $d_1 + d_2 = 2a$, where $d_1$ and $d_2$ are its distances to the two foci. This one fact solves many problems. If you also know an angle, the Law of Cosines gives a second equation.

In the triangle $F_1 P F_2$ the sides are $d_1$, $d_2$, and $2c$ (the distance between the foci). If $\theta$ is the angle at $F_1$, the Law of Cosines says $d_2^2 = d_1^2 + (2c)^2 - 2 d_1 (2c)\cos\theta$. Replace $d_2$ with $2a - d_1$ and the $d_1^2$ terms cancel, leaving a linear equation in $d_1$.

Building an equation from points and conditions

An ellipse through two given points

If the ellipse is centered at the origin with axes along the coordinate axes, write it as $\dfrac{x^2}{p} + \dfrac{y^2}{q} = 1$, or equivalently $Ax^2 + By^2 = 1$ with $A = \dfrac{1}{p}$ and $B = \dfrac{1}{q}$. Each point gives a linear equation in $A$ and $B$, so two points give a system you can solve by substitution or elimination.

Rectangles inscribed in an ellipse

A rectangle with sides parallel to the axes and its corners on the ellipse is symmetric about both axes. If one corner is $(x, y)$ with $x, y > 0$, the other corners are $(\pm x, \pm y)$. So the rectangle is $2x$ wide and $2y$ tall, and its area is $4xy$.

Tangent lines

A tangent line touches the ellipse at exactly one point. From the line-and-ellipse test, that means the discriminant is $0$. Doing the algebra once in general gives a handy shortcut.

This formula does not cover vertical lines, which have no slope. For this ellipse the vertical tangents are $x = \pm\sqrt{p}$.

If a problem asks for a parameter, like "for which $k$ is $x + 2y + k = 0$ tangent?", rewrite the line as $y = mx + (\text{intercept})$ first, and then use the condition on the intercept.

Focus and directrix

There is a second way to define an ellipse that does not use two foci at once.

More loci that give ellipses

Stretching or squashing a circle

If you keep every $x$-coordinate of a circle and multiply every $y$-coordinate by the same number $k$ with $0 < k < 1$, you squash the circle into an ellipse. Substitute the old coordinate (written in terms of the new one) into the circle's equation.

Circles tangent to a given circle

Suppose a circle has center $P$ and radius $r$, passes through a fixed point $A$, and is internally tangent to a big circle with center $O$ and radius $R$. Then $d(P, A) = r$, because $A$ is on the moving circle. Also $d(O, P) = R - r$, because the circles touch from the inside. Add them to eliminate $r$.

Back to the beginning: the string has fixed length, so the two pieces of string from the pencil to the tacks always add up to the same number. That is exactly the definition $d_1 + d_2 = 2a$. In a whispering gallery, every sound path from one focus to the wall and on to the other focus also has length $2a$. All the waves arrive together, and the whisper carries.

Summary