Hyperbola in a Cartesian Plane

Hyperbola in a Cartesian Plane

The set of all points in a plane such that the difference of the distances from two fixed points, called foci, is constant.

  • The midpoint of the segment that joins the foci is the centre of the hyperbola.
  • The line that passes through the two foci is the transverse axis and the line that passes through the centre, and that is perpendicular to the transverse axis, is the conjugate axis.
  • The basic equation of the relation that defines a hyperbola in a Cartesian plane is \(\dfrac {x^{2}} {a^{2}}−\dfrac {y^{2}} {b^{2}} = 1\) where a is the length of the semi-transverse axis and b is the length of the semi-conjugate axis.
  • The points with coordinates F1(c, 0) ou F2(−c, 0), where c² = a² + b², are the foci of the hyperbola.
  • The hyperbola is equilateral when a = b, that is, when the asymptotes are perpendicular.

  • A hyperbola can also be described as a section of a right circular cone that is obtained when a plane cuts the two nappes of the cone but does not pass through its vertex.


Etymological note

The term hyperbola is derived from two Greek words : υπερ (huper), meaning “beyond,” and βαλλω (ballô), meaning “to throw”.

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