In the definition of a circle, Definition circledefn, we fixed a point called the center and considered all of the points which were a fixed distance \(r\) from that one point. For our next conic section, the ellipse, we fix two distinct points and a distance \(d\) to use in our definition.

In the GeoGebra interactive below, we fix two points, \(F_{1}\) and \(F_{2}\), to serve as foci and fix a distance \(d = 4\) units. As we adjust the slider, we plot points whose distance to \(F_{1}\) plus the distance to \(F_{2}\) is exactly \(4\) units.

We can replicate this interactive physically by taking a length of string and anchoring it to two points on a piece of paper. The curve traced out by taking a pencil and moving it so the string is always taut is an ellipse.

Each ellipse has an assortment of parameters associated with it which we sketch below.

Figure 1: An ellipse with center \(C\); foci \(F_1\), \(F_2\); and vertices \(V_1\), \(V_2\)

As depicted above, the center of the ellipse is the midpoint of the line segment connecting the two foci. The major axis of the ellipse is the line segment connecting two opposite ends of the ellipse which also contains the center and foci. The minor axis of the ellipse is the line segment connecting two opposite ends of the ellipse which contains the center but is perpendicular to the major axis. The vertices of an ellipse are the points of the ellipse which lie on the major axis.

Notice that the center is also the midpoint of the major axis, hence it is the midpoint of the vertices. Also note that the major axis is the longer of the two axes through the center, hence the moniker ‘major.’ Likewise, the minor axis is the shorter of the two, whence the adjective ‘minor.’

In order to derive the standard equation of an ellipse, we assume that the ellipse has its center at \((0,0)\), its major axis along the \(x\)-axis, and has foci \((c,0)\) and \((-c,0)\) and vertices \((-a,0)\) and \((a,0)\). We will label the \(y\)-intercepts of the ellipse as \((0,b)\) and \((0,-b)\) (We assume \(a\), \(b\), and \(c\) are all positive numbers.)

Note that since \((a,0)\) is on the ellipse, it must satisfy the conditions of Definition ellipsedefn. That is, the distance from \((-c,0)\) to \((a,0)\) plus the distance from \((c,0)\) to \((a,0)\) must equal the fixed distance \(d\). Since all of these points lie on the \(x\)-axis, we get

\[ \begin{array}{rclr} \mbox {distance from $(-c,0)$ to $(a,0)$} + \mbox {distance from $(c,0)$ to $(a,0)$} & = & d & \\ (a+c) + (a-c) & = & d & \\ 2a & = & d \\ \end{array}\]

In other words, the fixed distance \(d\) mentioned in the definition of the ellipse is none other than the length of the major axis. We now use that fact \((0,b)\) is on the ellipse, along with the fact that \(d=2a\) to get

\[ \begin{array}{rclr} \mbox {distance from $(-c,0)$ to $(0,b)$} + \mbox {distance from $(c,0)$ to $(0,b)$} & = & 2a & \\ \sqrt {(0-(-c))^2+(b-0)^2} + \sqrt {(0-c)^2+(b-0)^2} & = & 2a & \\ \sqrt {b^2+c^2}+\sqrt {b^2+c^2} & = & 2a \\ 2 \sqrt {b^2+c^2} & = & 2a \\ \sqrt {b^2+c^2} & = & a \end{array}\]

From this, we get \(a^2 = b^2 + c^2\), or \(b^2 = a^2 - c^2\), which will prove useful later. Now consider a point \((x,y)\) on the ellipse. Applying Definition ellipsedefn, we get

\[ \begin{array}{rclr} \mbox {distance from $(-c,0)$ to $(x,y)$} + \mbox {distance from $(c,0)$ to $(x,y)$} & = & 2a & \\ \sqrt {(x-(-c))^2+(y-0)^2} + \sqrt {(x-c)^2+(y-0)^2} & = & 2a & \\ \sqrt {(x+c)^2+y^2}+\sqrt {(x-c)^2+y^2} & = & 2a \\ \end{array}\]

In order to make sense of this situation, we need to make good use of Intermediate Algebra.

\[ \begin{array}{rclr} \sqrt {(x+c)^2+y^2}+\sqrt {(x-c)^2+y^2} & = & 2a & \\ \sqrt {(x+c)^2+y^2} & = & 2a - \sqrt {(x-c)^2+y^2} & \\ \left (\sqrt {(x+c)^2+y^2}\right )^2 & = & \left (2a - \sqrt {(x-c)^2+y^2}\right )^2 & \\ (x+c)^2+y^2 & = & 4a^2 - 4a\sqrt {(x-c)^2+y^2} + (x-c)^2+y^2 & \\ 4a\sqrt {(x-c)^2+y^2} & = & 4a^2 + (x-c)^2 - (x+c)^2 & \\ 4a\sqrt {(x-c)^2+y^2} & = & 4a^2 - 4cx & \\ a\sqrt {(x-c)^2+y^2} & = & a^2 - cx & \\ \left (a\sqrt {(x-c)^2+y^2}\right )^2 & = & \left (a^2 - cx\right )^2 & \\ a^2\left ((x-c)^2+y^2\right ) & = & a^4 - 2a^2cx +c^2 x^2 & \\ a^2x^2 - 2a^2cx + a^2c^2+a^2 y^2 & = & a^4 - 2a^2cx +c^2 x^2 & \\ a^2x^2 - c^2 x^2 +a^2 y^2 & = & a^4 - a^2c^2 & \\ \left (a^2 - c^2\right ) x^2 +a^2 y^2 & = & a^2 \left (a^2 - c^2\right ) & \\ \end{array}\]

We are nearly finished. Recall that \(b^2 = a^2 - c^2\) so that

\[ \begin{array}{rclr} \left (a^2 - c^2\right )x^2 +a^2 y^2 & = & a^2\left (a^2 - c^2\right ) & \\ b^2 x^2 +a^2 y^2 & = & a^2 b^2 & \\ \frac {x^2}{a^2} + \frac {y^2}{b^2} & = & 1 & \\ \end{array}\]

This equation is for an ellipse centered at the origin. To get the formula for the ellipse centered at \((h,k)\), we could use the transformations from Section Transformations or re-derive the equation using Definition ellipsedefn and the distance formula to obtain the formula below.

Some remarks about Equation standardellipse are in order. First note that the values \(a\) and \(b\) determine how far in the \(x\) and \(y\) directions, respectively, one counts from the center to arrive at points on the ellipse.

Also note that if \(a > b\), then we have an ellipse whose major axis is horizontal, and hence, the foci lie to the left and right of the center. In this case, as we’ve seen in the derivation, the distance from the center to the focus, \(c\), can be found by \(c = \sqrt {a^2 - b^2}\).

If \(b > a\), the roles of the major and minor axes are reversed, and the foci lie above and below the center. In this case, \(c = \sqrt {b^2 - a^2}\). In either case, it’s best to just remember that \(c\) is the distance from the center to each focus, and, formulaically, \(c = \sqrt {\mbox {bigger denominator} - \mbox {smaller denominator}}\).

Finally, it is worth mentioning that if we compare Equation standardellipse with the alternate standard equation of the circle, Equation standardcirclealternate, the only difference between the forms is that with a circle, the denominators are the same, and with an ellipse, they are different.

If we take a transformational approach, we can consider both Equations standardellipse and standardcirclealternate as shifts and stretches of the Unit Circle \(x^2 + y^2 = 1\) in Definition UnitCircle. Replacing \(x\) with \((x-h)\) and \(y\) with \((y-k)\) causes the usual horizontal and vertical shifts. Replacing \(x\) with \(\frac {x}{a}\) and \(y\) with \(\frac {y}{b}\) causes the usual vertical and horizontal stretches.

In other words, it is perfectly fine to think of an ellipse as the deformation of a circle in which the circle is stretched farther in one direction than the other.

As seen in Example ellipsesfirstex above, it is often necessary to algebraically manipulate a given equation into the standard form of Equation standardellipse in order to graph. We summarize one approach below.

To Write the Equation of an Ellipse in Standard Form
  1. Group common variables together on one side of the equation and put the constant on the other.
  2. Complete the square on both variables as needed.
  3. Divide both sides, if needed, to obtain \(1\) on one side of the equation.

If we think of a circle as being ‘perfectly round,’ then ellipses, being deformed circles, have varying degrees of ‘roundness.’ We quantify this idea with the notion of eccentricity defined formally below.

In an ellipse, the foci are closer to the center than the vertices, so \(0 < e < 1\). Using the GeoGebra interactive below, we can adjust the distances from the foci to the center to produce a wide range of ellipses and their corresponding eccentricities. In general, the closer the eccentricity is to \(0\), the less ‘eccentric’ or more ‘circular’ the ellipse appears. On the other hand, the closer the eccentricity is to \(1\), the more ‘eccentric’ the ellipse is and it appears less ‘circular.’

According to Kepler’s Laws of Planetary Motion, each planet orbits the Sun in an elliptical path with the Sun at one focus. The eccentricity is therefore an important orbital parameter. We investigate the orbit of Mercury in the following example.

As with parabolas, ellipses have a reflective property. If we imagine the dashed lines below representing sound waves, then it can be shown that the waves emanating from one focus reflect off the top of the ellipse and head towards the other focus.

Such geometry is exploited in the construction of so-called ‘Whispering Galleries’. If a person whispers at one focus, a person standing at the other focus will hear the first person as if they were standing right next to them. We explore the Whispering Galleries in our last example.