The ellipse

What you will be able to do

• draw an ellipse with two pegs and a loop of string, and explain why it works;
• read the axes, foci, directrices, eccentricity and latus rectum from the equation;
• use the auxiliary circle and the parametric form;
• write the equation of an ellipse from its foci and one more fact.

A gardener and a loop of string

A gardener wants an oval flower bed. She pushes two pegs into the soil, ties a loop of string loosely round both, and draws the loop tight with a stick. Then she moves the stick all round, keeping the string tight. The stick draws an ellipse.

Why? The string has a fixed length. Its two straight parts run from the stick to the two pegs, and the part between the pegs is fixed. So the sum of the two distances from the stick to the pegs is constant. That is the definition of an ellipse.

Two definitions that say the same

By two foci. Let $F_1$ and $F_2$ be two points, the foci, with $F_1F_2 = 2c$. The ellipse is the set of points $P$ with

By focus and directrix. For a point $F$ and a line $l$, the ellipse is the set of points with $PF = e \cdot PM$, where $PM$ is the distance to the line and $0 < e < 1$. This is the same rule as for the parabola, with $e = 1$ replaced by a smaller ratio. An ellipse has two foci and two directrices, and either pair gives the whole curve.

The two definitions describe the same curve, and $c = ae$.

The standard equation

Put the centre at the origin and the foci on the $x$-axis at $(\pm c, 0)$. Write $PF_1 + PF_2 = 2a$ and square twice. The result is

Here is the vocabulary.

• Major axis: the long axis, along the $x$-axis, of length $2a$. Its ends $(\pm a, 0)$ are the vertices.
• Minor axis: the short axis, along the $y$-axis, of length $2b$. Its ends are $(0, \pm b)$.
• Foci: $(\pm ae, 0)$, which is $(\pm c, 0)$ with $c = \sqrt{a^2 - b^2}$.
• Eccentricity: $e = \dfrac{c}{a} = \sqrt{1 - \dfrac{b^2}{a^2}}$. It is between $0$ and $1$. At $e = 0$ the ellipse is a circle.
• Directrices: $x = \pm\dfrac{a}{e}$.
• Latus rectum: the chord through a focus perpendicular to the major axis. Its length is $\dfrac{2b^2}{a}$. The ends are $\left(\pm ae, \pm\dfrac{b^2}{a}\right)$.
• Focal distances: a point $(x_1, y_1)$ has $PF_1 = a + ex_1$ and $PF_2 = a - ex_1$. They add up to $2a$.

If the larger number is under $y^2$, so that $\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1$ with $a > b$, the major axis is along the $y$-axis and the foci are $(0, \pm ae)$, the directrices are $y = \pm\dfrac{a}{e}$.

A worked example

Take $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$, so $a = 5$ and $b = 3$.

1. The focal length. $c = \sqrt{25 - 9} = 4$. The foci are $(\pm 4, 0)$.
2. The eccentricity. $e = \dfrac{4}{5} = 0.8$.
3. The directrices. $x = \pm\dfrac{a}{e} = \pm\dfrac{25}{4} = \pm 6.25$.
4. The latus rectum. $\dfrac{2b^2}{a} = \dfrac{18}{5} = 3.6$, with ends $(4, \pm 1.8)$.
5. A check. The point $P(4, 1.8)$ has $PF_2 = 1.8$, and its distance from the directrix $x = 6.25$ is $2.25$. The ratio is $\dfrac{1.8}{2.25} = 0.8 = e$.

Next take $9x^2 + 16y^2 = 144$. Divide by $144$: $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. Then $a = 4$, $b = 3$, $c = \sqrt{7} \approx 2.646$ and $e = \dfrac{\sqrt{7}}{4} \approx 0.661$.

Now take $4x^2 + y^2 = 16$, which is $\dfrac{x^2}{4} + \dfrac{y^2}{16} = 1$. The larger number $16$ is under $y^2$, so $a = 4$ and $b = 2$, and the major axis is along the $y$-axis. Then $c = \sqrt{16 - 4} = 2\sqrt{3}$, the foci are $(0, \pm 2\sqrt{3})$ and $e = \dfrac{\sqrt{3}}{2}$.

Finding the equation

An ellipse has foci $(\pm 3, 0)$ and eccentricity $\dfrac{3}{5}$. Then $c = 3$ and $a = \dfrac{c}{e} = 5$. So $b^2 = 25 - 9 = 16$ and the ellipse is $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$.

Always find $a$ and $c$ first, then $b^2 = a^2 - c^2$. The most common slip is to use $b^2 = a^2 + c^2$, which belongs to the hyperbola.

The auxiliary circle and the parametric form

The circle $x^2 + y^2 = a^2$ with the major axis as its diameter is the auxiliary circle. Take the point $Q(a\cos\theta, a\sin\theta)$ on it. Drop a perpendicular from $Q$ to the major axis. It meets the ellipse at the point

Check: $\dfrac{a^2\cos^2\theta}{a^2} + \dfrac{b^2\sin^2\theta}{b^2} = 1$. The angle $\theta$ is the eccentric angle of $P$. Every point of the ellipse has this form. Compare the circle from the first course, $x = r\cos\theta$, $y = r\sin\theta$. The ellipse is the circle squeezed in the $y$-direction by the factor $\dfrac{b}{a}$. That is why its area is $\pi ab$, the result you used in Applications of Integrals.

For $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ and $\theta = 60^\circ$ the point is $(5\cos 60^\circ, 3\sin 60^\circ) = (2.5, 2.598)$.

Practise with new numbers

Try it in Python

The program takes an ellipse and finds its foci, eccentricity, directrices and latus rectum. It checks that $PF_1 + PF_2 = 2a$ at several points and that $PF_2 = e \times$ the distance from the directrix.

Common mistakes

• Using $b^2 = a^2 + c^2$. For an ellipse $c^2 = a^2 - b^2$. The semi-major axis $a$ is always the largest.
• Taking $a$ to be the number under $x^2$. The number under $y^2$ may be larger.
• Forgetting that $e < 1$. If your $e$ is larger than $1$, you have made a slip.
• Mixing the axes and the semi-axes. The major axis has length $2a$, not $a$.

Remember these

• $PF_1 + PF_2 = 2a$, or $PF = e \cdot PM$ with $0 < e < 1$.
• $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with $b^2 = a^2(1 - e^2)$. Foci $(\pm ae, 0)$. Directrices $x = \pm\dfrac{a}{e}$.
• Latus rectum $\dfrac{2b^2}{a}$. Focal distances $a \pm ex_1$.
• The point $(a\cos\theta, b\sin\theta)$, with the auxiliary circle $x^2 + y^2 = a^2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the eccentricity of $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$.

2. Find the foci of $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$.

3. Find the length of the latus rectum of $\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1$.

4. An ellipse has foci $(\pm 5, 0)$ and $a = 13$. Find its equation.

5. What is the sum of the focal distances of any point on $\dfrac{x^2}{49} + \dfrac{y^2}{24} = 1$?

References

• NCERT, Mathematics for Class XI, Chapter 10 (Conic Sections), the section on the ellipse.
• OpenStax, Precalculus 2e, Chapter 12 (Analytic Geometry), the section on the ellipse.