The ellipse as a set of points, its standard equation with a horizontal or vertical major axis, vertices, co-vertices, foci, eccentricity and ellipses centered at any point.
The definition
Pin the two ends of a string to a board, pull it tight with a pencil and trace all the way around. The curve is an ellipse: the shape of planetary orbits.
Standard equations
For an ellipse centered at the origin, let \(a\) be the distance from the center to a vertex (the ends of the long axis) and \(b\) the distance to a co-vertex (the ends of the short axis), with \(a > b\). The foci are at distance \(c\) from the center, where
\[c^2 = a^2 - b^2\]
The larger denominator is always \(a^2\), and it tells you the direction of the major axis.
Eccentricity
When \(e\) is close to \(0\) the foci are near the center and the ellipse is nearly a circle (a circle has \(e = 0\)). When \(e\) is close to \(1\) the ellipse is long and thin. Earth's orbit has \(e \approx 0.017\): almost a circle.
Ellipses centered at \((h, k)\)
Replace \(x\) with \(x - h\) and \(y\) with \(y - k\) to move the center to \((h, k)\).
Practice
Quick check
For an ellipse with \(a > b\), how are \(a\),\(b\) and \(c\) related?
The foci are inside the ellipse, so \(c < a\):\(c^2 = a^2 - b^2\).
Quick check
An ellipse with eccentricity \(0\) is:
\(e = 0\) means \(c = 0\): both foci are at the center, which makes a circle.
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