Ellipse Equation, Standard Form, and Foci

An ellipse is one of the four conic sections, alongside the circle, parabola, and hyperbola. It looks like a stretched circle, and its equation shows up anywhere something orbits or curves symmetrically, from planetary orbits to whispering-gallery ceilings. This lesson covers the standard form of the ellipse equation, how to find its foci and vertices, and how it connects to the other conics.

An ellipse is the set of all points in a plane where the sum of the distances to two fixed points, called the foci (singular: focus), is always the same constant value. If you tied a string between two pins and traced a pencil around the taut string, the shape you would draw is an ellipse. This definition can be turned into an equation using the distance formula to measure the distance from any point \((x, y)\) on the curve to each focus.

For an ellipse centered at the origin, the standard form depends on whether the major axis (the longer axis) is horizontal or vertical.

Horizontal major axis: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a > b\).

Vertical major axis: \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \(a > b\) again, but now \(a^2\) sits under the \(y^2\) term. This is the equation of a vertical ellipse.

Here \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis. If the ellipse is shifted so its center is at \((h, k)\) instead of the origin, the equation becomes \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) (or with \(a^2\) and \(b^2\) swapped for a vertical ellipse).

The vertices are the endpoints of the major axis, and the co-vertices are the endpoints of the minor axis. The foci lie on the major axis, a distance \(c\) from the center, where:

\( c^2 = a^2 - b^2 \)

For a horizontal ellipse, the foci are at \((\pm c, 0)\); for a vertical ellipse, they are at \((0, \pm c)\). The diagram below labels each of these key parts.

An ellipse centered at the origin can also be described parametrically as \(x = a \cos t\) and \(y = b \sin t\), where \(t\) runs from \(0\) to \(2\pi\). Substituting these into \( \frac{x^2}{a^2} + \frac{y^2}{b^2} \) gives \( \cos^2 t + \sin^2 t \), which always equals \(1\), confirming the point lies on the ellipse for every value of \(t\).

Write \( 9x^2 + 25y^2 = 225 \) in standard form and identify its key features.

Divide every term by \(225\): \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \).

So \(a^2 = 25\), giving \(a = 5\), and \(b^2 = 9\), giving \(b = 3\). Since \(25 > 9\), the major axis is horizontal.

Vertices: \((\pm 5, 0)\). Co-vertices: \((0, \pm 3)\).

Foci: \( c^2 = a^2 - b^2 = 25 - 9 = 16 \), so \(c = 4\), and the foci are at \((\pm 4, 0)\).

The eccentricity of an ellipse is \( e = \frac{c}{a} \), a value between \(0\) and \(1\) that measures how stretched the ellipse is (an eccentricity near \(0\) is almost circular, while a value close to \(1\) is very elongated). Each ellipse has two directrices, vertical or horizontal lines given by \( x = \pm \frac{a}{e} \) (or \( y = \pm \frac{a}{e} \) for a vertical ellipse). Every point on the ellipse satisfies a fixed ratio between its distance to a focus and its distance to the corresponding directrix, and that ratio is exactly the eccentricity.

When \(a = b\), the ellipse equation reduces to \( x^2 + y^2 = r^2 \), which is just a circle, so a circle is really a special case of an ellipse. By contrast, a hyperbola has a minus sign between the squared terms instead of a plus sign, which produces two separate branches instead of one closed curve. Recognizing this pattern of signs and squared terms is the fastest way to tell the conic sections apart.

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