# Hyperbola - Other Definitions, Features, Drawing Hyperbolas, Uses

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plane cone nappes curve

A hyperbola is a **curve** formed by the intersection of a right circular cone and a **plane** (see Figure 1). When the plane cuts both nappes of the cone, the intersection is a hyperbola. Because the plane is cutting two nappes, the curve it forms has two U-shaped branches opening in opposite directions.

## Additional Topics

A hyperbola can be defined in several other ways, all of them mathematically equivalent:
Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group.
Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.
Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group.
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A hyperbola is symmetric about both its transverse and its conjugate axes.
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Hyperbolas can be sketched quite accurately by first locating the vertices, the foci, and the asymptotes. Starting with the axes, locate the vertices and foci. Draw a circle with its center at C, passing through the two foci. Draw lines through the vertices perpendicular to the transverse axis. This determines four points, which are corners of a rectangle. These diagonals are the asymptotes. Using…

Hyperbolas have many uses, both mathematical and practical. The hyperbola y = 1/x is sometimes used in the definition of the natural logarithm. In Figure 6 the logarithm of a number n is represented by the shaded area, that is, by the area bounded by the x-axis, the line x = 1, the line x = n, and the hyperbola. Of course one needs calculus to compute this area, but there are techniques for doing …

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