Vertex of Hyperbola
Vertex of hyperbola is a point where the hyperbola cuts the axis of the hyperbola. The hyperbola cuts the axis at two distinct points, and hence the hyperbola has two vertices. The midpoint of the vertex is the center of the hyperbola, and the vertex of hyperbola, the foci of hyperbola are collinear.
Let us learn more about the vertex of the hyperbola, its properties and related terms, with the help of examples, FAQs.
1.  What Is Vertex of Hyperbola? 
2.  Properties of Vertex of Hyperbola 
3.  Terms Related to Vertex of Hyperbola 
4.  Examples on Vertex of Hyperbola 
5.  Practice Questions 
6.  FAQs on Vertex of Hyperbola 
What Is Vertex of Hyperbola?
Vertex of hyperbola is a point where the hyperbola has the maximum turn. Vertex of hyperbola is the point where the axis of the hyperbola cuts the hyperbola. The hyperbola cuts the axis at two distinct points which are the vertices of the hyperbola. The vertex of the hyperbola and the foci of hyperbola are collinear and lie on the axis of the hyperbola.
Equation of Hyperbola: \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\)
Vertices of Hyperbola: (a, 0), and (a, 0)
For a hyperbola, the vertices are equidistant from the center of the hyperbola, and the distance between the two vertices of the hyperbola are 2a units.
Properties of Vertex of Hyperbola
The following properties of vertex of hyperbola help in a better understanding of the vertex of the hyperbola.
 The hyperbola has two vertices.
 The vertex of the hyperbola is represented as coordinates of a point in the twodimensional coordinate axis.
 The vertex of the hyperbola lies on the axis of the hyperbola.
 The midpoint of vertices of the hyperbola is the center of the hyperbola.
 The distance between the vertices of the hyperbola is equal to '2a' units.
Terms Related to Vertex of Hyperbola
The following concepts help in an easier understanding of the vertex of the hyperbola.
 Foci of Hyperbola: The hyperbola has two foci, and for the hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\), the two foci are (+ae, 0), and (ae, 0). The two foci are equidistant from the center of the hyperbola.
 Directrix of Hyperbola: The directrix of a hyperbola is a line parallel to the latus rectum of the hyperbola, and is perpendicular to the axis of the hyperbola. For a hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\), the directric is x = +a/e, and x = a/e.
 Latus Rectum of Hyperbola: The line passing through the foci of the hyperbola and perpendicular to the axis of the hyperbola is the latus rectum, The hyperbola has two foci, and hence has two latus rectums. For a hyperbola, \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) the length of the latus rectum is 2b^{2}/a.
 Axis of Hyperbola: The line passing through the foci and the center of the hyperbola is the axis of the hyperbola. The latus rectum and the directrix are perpendicular to the axis of the hyperbola. For a hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) the xaxis is the axis of hyperbola and has the equation y = 0.
 Eccentricity of Hyperbola: The eccentricity of the hyperbola refers to how curved the conic is. For a hyperbola, the eccentricity is greater than 1 (e > 1). The formula of eccentricity of a hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) is \(e = \sqrt {1 + \dfrac{b^2}{a^2}}\).
 Rectangular Hyperbola: The hyperbola having both the major axis and minor axis of equal length is called a rectangular hyperbola. Here we have 2a = 2b, and the equation of rectangular hyperbola is \(\dfrac{x^2}{a^2}  \dfrac{y^2}{a^2} = 1\).
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Examples on Vertex of Hyperbola

Example 1: Find the vertex of hyperbola having the equation \(\dfrac{x^2}{25}  \dfrac{y^2}{16}=1\).
Solution:
The given equation of hyperbola is \(\dfrac{x^2}{25}  \dfrac{y^2}{16}=1\).
Comparing this with the standard equation of hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2}=1\) we gave \(a^2 = 25\), and \(b^2 = 16\)
Hence we have a = 5, and b = 4.
The required vertex of hyperbola is (+a, 0) = (+5, 0), and (a, 0) = (5, 0).
Therefore the vertices of the hyperbola are (+5, 0), and (5, 0).

Example 2: Find the equation of the hyperbola having the vertices (+4, 0), and the eccentricity of 3/2.
Solution:
The given vertex of hyperbola is (a, 0) = (4, 0), and hence we have a = 4.
The eccentricity of the hyperbola is e = 3/2
Let us find the length of the semiminor axis 'b', with the help of the following formula.
\(b^2 = a^2(e^2  1)\)
\(b^2 = 4^2((3/2)^2  1)\)
\(b^2 = 16(9/4  1)\)
\(b^2 = 16 × (9  4)/4 \)
\(b^2 = 16 × 5/4\)
\(b^2 = 20\)
The required equation of hyperbola is \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\)
\(\dfrac{x^2}{16}  \dfrac{y^2}{20} = 1 \)
Therefore the required equation of hyperbola is \(\dfrac{x^2}{16}  \dfrac{y^2}{20} = 1 \).
FAQs on Vertex of Hyperbola
What Is The Vertex Of Hyperbola In Geometry?
The points were the hyperbola takes the maximum curve is called the vertex of hyperbola. There are two vertex of hyperbola and they lie on the major axis of the hyperbola. The equation of hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) has two vertices (+a, 0), and (a, 0).
How to Know If a Point Is A Vertex Of Hyperbola?
The two points can be identified as the vertices of the hyperbola if it satisfies the equation of the hyperbola. The two vertices represented as points should satisfy the equation of hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\).
How To Find Vertex Of Hyperbola From The Equation Of Hyperbola?
The vertex of hyperbola can be found from the equation of hyperbola in two simple steps. From the equation of hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\), we can find the value of 'a'. And the two vertices of the hyperbola are (+a, 0), and (a, 0).
How Many Vertex Does A Hyperbola Have?
The hyperbola cuts the major axis at two distinct points and hence it has two vertices. The given equation of hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) has the two vertices (+a, 0), and (a, 0).
What Are The Application Of Vertex Of Hyperbola?
The vertex of hyperbola is helpful to find the equation of hyperbola, the eccentricity of hyperbola, and also the length of major axis of the hyperbola.
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