Ellipses, calculus lesson

Tie a loop of string around two pins on a board, pull it tight with a pencil and move the pencil around. The curve you draw is an ellipse, and the pins are its foci. Planets move on ellipses with the sun at one focus, so this shape is worth knowing well.

Key terms

Sum of focal distances

Definition Let $F_1$ and $F_2$ be two points with $\lvert F_1F_2 \rvert = 2c$ , $c \ge 0$ , and let $a > c$ . The ellipse with foci $F_1$ and $F_2$ is the set of points $P$ with

\lvert PF_1 \rvert + \lvert PF_2 \rvert = 2a.

The condition $2a > 2c$ is needed. If $2a = 2c > 0$ the set is just the segment $F_1F_2$ , and if $2a < 2c$ it is empty.

In plain words The total distance from a point on the ellipse to the two foci is always the same. That total is the length of the string in the pin construction.

Example Take foci $(\pm 3, 0)$ and $2a = 10$ . The point $(0, 4)$ is $5$ from each focus, and $5 + 5 = 10$ , so it lies on the ellipse.

Foci and the standard equation

Definition With foci $(\pm c, 0)$ and $a > c \ge 0$ , set $b = \sqrt{a^2 - c^2}$ , so $a \ge b > 0$ . The ellipse is

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad c^2 = a^2 - b^2.

With foci $(0, \pm c)$ the roles of $x$ and $y$ swap, giving

\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1.

In either form the foci lie on the longer axis, and $a$ is the larger of the two denominators’ square roots.

In plain words The foci sit inside the ellipse on its long axis, at distance $c$ from the center, and you find $c$ from $c^2 = a^2 - b^2$ .

Example For $\frac{x^2}{9} + \frac{y^2}{5} = 1$ we have $a = 3$ , $b = \sqrt{5}$ and $c = \sqrt{9 - 5} = 2$ . The foci are $(\pm 2, 0)$ . With foci $(\pm 3, 0)$ and $a = 5$ from the previous example, $b^2 = 25 - 9 = 16$ , so that ellipse is

\frac{x^2}{25} + \frac{y^2}{16} = 1.

Major and minor axes

Definition For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$ , the major axis is the segment from $(-a, 0)$ to $(a, 0)$ , of length $2a$ , and it contains the foci. The minor axis is the segment from $(0, -b)$ to $(0, b)$ , of length $2b$ . The numbers $a$ and $b$ are the semimajor and semiminor axes, and the axes meet at the center.

In plain words The major axis is the longest line through the middle of the ellipse, and the minor axis is the shortest.

Example Dividing $4x^2 + y^2 = 36$ by $36$ gives

\frac{x^2}{9} + \frac{y^2}{36} = 1.

Now the larger denominator is under $y^2$ , so the major axis is vertical with length $12$ and the minor axis is horizontal with length $6$ . Here $c = \sqrt{36 - 9} = 3\sqrt{3}$ and the foci are $(0, \pm 3\sqrt{3})$ .

Vertices

Definition The vertices of an ellipse are the two endpoints of its major axis. For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$ they are $(\pm a, 0)$ . The endpoints $(0, \pm b)$ of the minor axis are often called the covertices.

In plain words The vertices are the two far ends of the ellipse along its long direction.

Example The ellipse $\frac{x^2}{9} + \frac{y^2}{36} = 1$ has vertices $(0, \pm 6)$ and covertices $(\pm 3, 0)$ .

Eccentricity

Definition The eccentricity of the ellipse with semimajor axis $a$ and focal distance $c$ is

e = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}, \qquad 0 \le e < 1.

In plain words It tells how stretched the ellipse is. Near $0$ it is almost a circle, near $1$ it is long and thin.

Example For $\frac{x^2}{25} + \frac{y^2}{16} = 1$ , $c = 3$ and $a = 5$ , so $e = \frac{3}{5}$ .

Directrices

Definition For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$ and $e = \frac{c}{a}$ , the directrices are the lines

x = \pm \frac{a}{e} = \pm \frac{a^2}{c}.

Every point $P$ on the ellipse satisfies $\lvert PF \rvert = e \cdot \operatorname{dist}(P, L)$ , where $F = (c, 0)$ goes with $L$ the line $x = \frac{a}{e}$ , and $F = (-c, 0)$ goes with $x = -\frac{a}{e}$ . This matches the focus directrix definition with $0 < e < 1$ .

In plain words Each focus has a partner line outside the ellipse. The distance to the focus is always the same fraction $e$ of the distance to that line.

Example For $\frac{x^2}{25} + \frac{y^2}{16} = 1$ the directrices are $x = \pm \frac{25}{3}$ . Check the point $(0, 4)$ with $F = (3, 0)$ . Its distance to $F$ is $5$ , its distance to $x = \frac{25}{3}$ is $\frac{25}{3}$ , and $\frac{3}{5} \cdot \frac{25}{3} = 5$ .

Reflection property

Definition Let $P$ be a point on an ellipse with foci $F_1$ and $F_2$ . The tangent line at $P$ makes equal angles with the segments $PF_1$ and $PF_2$ . Equivalently, the normal line at $P$ bisects the angle $F_1PF_2$ . So a ray from one focus that reflects off the inside of the ellipse passes through the other focus.

In plain words In an elliptical room, a whisper at one focus bounces off the walls and arrives at the other focus.

Example Take $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and $P = \left(3, \frac{16}{5}\right)$ . The tangent is $\frac{3x}{25} + \frac{y}{5} = 1$ , with normal direction $(3, 5)$ . The vectors from $P$ to the foci are $(-6, -3.2)$ and $(0, -3.2)$ , with lengths $6.8$ and $3.2$ (and $6.8 + 3.2 = 10$ ). The cosines of their angles with $(-3, -5)$ are

\frac{18 + 16}{6.8\sqrt{34}} = \frac{5}{\sqrt{34}}, \qquad \frac{0 + 16}{3.2\sqrt{34}} = \frac{5}{\sqrt{34}},

so the normal splits the angle into two equal parts.

Circle as a special case

Definition When $a = b$ , the equation becomes $x^2 + y^2 = a^2$ . Then $c = 0$ , the two foci coincide at the center, $e = 0$ , and the sum condition reads $2\lvert PC \rvert = 2a$ , which is the definition of the circle of radius $a$ about the center $C$ . A circle has no directrix.

In plain words A circle is an ellipse whose two foci have moved together into the center.

Example The circle $x^2 + y^2 = 9$ is the ellipse with $a = b = 3$ . Every point on it is $3$ from the center, so its distances to the two coinciding foci add to $6 = 2a$ .

Check yourself

Find the foci and eccentricity of $\frac{x^2}{169} + \frac{y^2}{144} = 1$ .

Show the answer

Here $a = 13$ and $c = \sqrt{169 - 144} = 5$ , so the foci are $(\pm 5, 0)$ and $e = \frac{5}{13}$ .

An ellipse has vertices $(0, \pm 5)$ and foci $(0, \pm 4)$ . Write its equation.

Show the answer

The major axis is vertical with $a = 5$ and $c = 4$ , so $b^2 = 25 - 16 = 9$ , and the equation is

\frac{x^2}{9} + \frac{y^2}{25} = 1.

An ellipse centered at the origin has its major axis on the $x$ axis, $a = 10$ and $e = 0.6$ . Find its directrices.

Show the answer

The directrices are the lines

x = \pm \frac{a}{e} = \pm \frac{10}{0.6} = \pm \frac{50}{3}.

Ellipses

Key terms

Sum of focal distances

Foci and the standard equation

Major and minor axes

Vertices

Eccentricity

Directrices

Reflection property

Circle as a special case

Check yourself

How this lesson connects

Builds on

Used later in