Vertex of a Hyperbola

 A hyperbola is defined as a smooth curve which tends to lie down in the plane and it can be described by the geometric properties or by the type of equation for which it may be the solution set. A hyperbola can be defined in the form of a curve that is represented by the function f(x) = 1 / x in the Cartesian plane.

In this article we are going to see about the vertex of a hyperbola.

 

How to Find the Vertex of a Hyperbola:

 

  •  In the case of ellipse, there exists a relationship between x, y, and z axes with ellipses, the computations are long and painful. So that, for hyperbolas (where x < z), the relationship is z2 – x2 = y2 or, which means the same thing, z2 = y2 + x2.
  •  This has been proved using the Pythagorean Theorem.
  • When the transverse axis is in the horizontal plane (when the center, and vertex line up side by side which is parallel to the horizontal axis), then the x2 goes with the part of the hyperbola's equation, and the y part is subtracted.

                                          ((a – y) 2 / x2) – ((b – c) 2 / y2) = 1.

 

Example to find the vertex of a hyperbola:

 

Ex 1: Given a hyperbola (x2 / 25) – (y/ 9) = 1. Find the vertex of the given hyperbola.

Sol :  The given hyperbola opens right or left because it is of the form x – y

        Here a2 = 25, b2 = 9, c2 = 25+9 = 34.

              Hence a = 5, b = 3 and c = 5.8

The vertex of a hyperbola is given by (a, 0), (-a, 0)

         Thus the vertex is (5, 0), (-5, 0).

 

Ex 2 : Given a hyperbola (y2 / 16) – (x/ 9) = 1. Find the vertex and foci of the given hyperbola.

Sol :  The hyperbola opens up and down because it is of the form y – x.

Here a2 = 16, b2 = 9, c2 = 16 + 9 = 25.

               a = 4, b = 3, c = 5

The vertex of a hyperbola are given by (0, a), (0,-a)

 Vertex of a hyperbola: (0, 4) and (0,-4).