Define rectangular hyperbola and find its eccentricity.

A rectangular hyperbola is a specific type of hyperbola where the transverse and conjugate axes are equal in length. This means that the asymptotes of the hyperbola are perpendicular to each other, forming a right angle. The standard equation of a rectangular hyperbola can be expressed as:

Standard Equation

The equation is typically written as:

• xy = c² (where c is a constant)

Characteristics of Rectangular Hyperbola

Some key features include:

• Asymptotes intersect at right angles.
• Symmetrical about the origin.

Eccentricity of a Rectangular Hyperbola

The eccentricity (e) of a rectangular hyperbola is a measure of how much it deviates from being circular. For a rectangular hyperbola, the eccentricity is always:

• e = √2

This value indicates that the rectangular hyperbola is more elongated than a circle, reflecting its unique geometric properties.