How Is the Transverse Axis of a Hyperbola Found?

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The transverse axis of a hyperbola passes through the center and the vertices of both hyperbolic curves. When extended further, the transverse axis also intersects the foci of both curves. The length of a transverse axis for any hyperbola with a given algebraic equation [(x-h)^2]/(a^2) - [(y-k)^2]/(b^2) = 1 is 2a.

The algebraic equation representing a hyperbola should be arranged such that it follows the format [(x-h)^2]/(a^2) - [(y-k)^2]/(b^2) = 1. If the arrangement conforms to this format, then the transverse axis is horizontal. Otherwise, if the arrangement produces the form [(y-k)^2]/(b^2) - [(x-h)^2]/(a^2) = 1, then the transverse axis is vertical.

The length of the transverse axis can be calculated as 2a, in which the value of the variable a can be extracted from the rearranged algebraic equation. To determine the endpoints of the transverse axis, which is actually the vertices of the two hyperbolic curves, the center must be identified at the set of coordinates (h,k). From this point, the two vertices may be determined as (h+a,k) and (h-a,k) for a hyperbola with horizontal transverse axis. Meanwhile, the endpoints of a vertical transverse axis are (h,k+a) and (h,k-a).

The hyperbola belongs to a special class of algebraic curves known as conic sections. The hyperbola is produced from two right circular cones that meet at the apex and intersect with a plane.