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.
((a – y)^{ 2} / x^{2}) – ((b – c)^{ 2} / y^{2}) = 1.
Ex 1: Given a hyperbola (x^{2} / 25) – (y^{2 }/ 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 a^{2} = 25, b^{2} = 9, c^{2} = 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 (y^{2} / 16) – (x^{2 }/ 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 a^{2} = 16, b^{2} = 9, c^{2} = 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).