Hyperbola Constructions

Solutions for the six hyperbola challenges can be found at the links below.

Hyperbola 1
Hyperbola 2
Hyperbola 3
Hyperbola 4
Hyperbola 5
Hyperbola 6

The hyperbola has many properties closely corresponding to those of the ellipse. Unlike the ellipse, it is intersected by only one of its axes, which often helps to simplify the construction. Also, this is the only conic section having asymptotes. The figure below shows everything I would wish to see in a completed hyperbola construction. That includes the center, both axes, both axial vertices, both foci, both directrices, both asymptotes, and, of course, the curve itself.

The line through the foci is the transverse axis, though the same term often is used for the line segment connecting the axial vertices. The conjugate axis has no vertices, but it too is often referred to as having finite length. There are reasons for that having little to do with these constructions, except in a certain sense shown below. Through both of the axial vertices construct lines perpendicular to the transverse axis, and let them intersect the asymptotes at the vertices of a rectangle. The side length of the rectangle in the ordinate direction is said to be the length of the conjugate axis. Here the conjugate axis is trimmed to fit that length. Half of that length might be called the conjugate radius.

Now construct a line perpendicular to the transverse axis, with its distance from the center equal to the conjugate radius. Equivalently, in the figure above, we could take one of the horizontal sides of the rectangle, rotate it 90° about the center, and produce it in both directions. Though it is useful for these constructions, to my knowledge this line is not widely embraced as a fundamental component of the hyperbola with a name of its own. Here let it be called line k.

Here is how I find line k useful in hyperbola constructions. Most of these objects will be hidden now. Let O be the center of the hyperbola and center of a circle with its diameter defined by the two axial vertices. Construct line OP, where P is a point on the circle. Construct PQ, perpendicular to OP, meeting the transverse axis at Q. Let R be the intersection of OP and line k. Now construct S, so that QS is perpendicular to the transverse axis and RS is parallel to the transverse axis. The hyperbola is the locus of S as P travels on its circle path.

This locus construction lies beneath all of the hyperbolas shown in this section of the Conic Challenges.

Back to Conic Challenges

Last update: April 19, 2026 ... Paul Kunkel whistling@whistleralley.com
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