Hyperbolen

A hyperbola forms when a plane cuts through both nappes of a cone, creating two open curves called branches.

The branches extend along the transverse axis of length 2a, where a is the distance from the center to each vertex.

Perpendicular to this lies the conjugate axis, with length 2b, defining a rectangle with dimensions 2a by 2b, whose diagonals extend outward as asymptotes that guide but never intersect the branches.

A hyperbola is defined as the set of points where the absolute difference in distances to two fixed points, called foci, is constant and equal to 2a.

The foci are placed along the x-axis at minus c and plus c, where c is the distance from the center to each focus.

Applying the distance formula between point P and each focus leads to expressions that, when squared, remove the square roots. The squared term is then expanded, followed by algebraic simplifications.

Further squaring and simplifying eliminates the remaining radical. Then, substituting the relation b squared equals c squared minus a squared — a form of the Pythagorean Theorem — gives the standard equation.

Hyperbolic shapes are used in cooling towers because their shape enhances strength and airflow.