Foci of Hyperbola
Foci of hyperbola are the two points on the axis of hyperbola and are equidistant from the center of the hyperbola. For the hyperbola the foci of hyperbola and the vertices of hyperbola are collinear. The eccentricity of hyperbola is defined with reference to the foci of hyperbola.
Let us learn more about the foci of hyperbola, its properties, terms related to it, with the help of examples, FAQs.
What Is Foci of Hyperbola?
Foci of hyperbola are points on the axis of hyperbola. For the hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) the two foci are (+ae, 0), and (ae, 0). The hyperbola is defined with reference to the foci of hyperbola, and for any point on the hyperbola, the ratio of its distance from the foci and its distance from the directrix is a constant value called the eccentricity of hyperbola and is greater than 1. (e > 1).
The midpoint of the foci of the hyperbola is the center of the hyperbola. The foci of the hyperbola and the vertices of the hyperbola are collinear and lie on the axis of the hyperbola.
Foci of Hyperbola Formulas
We have already seen that the foci of a hyperbola that is of the form \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) are given by (± ae, 0), where 'e' is the eccentricity of the hyperbola. But the formula for foci depends upon the type of the hyperbola. The formulas are tabulated below.
Hyperbola  Foci 

x^{2}/a^{2}  y^{2}/b^{2} = 1  (± ae, 0), where e = \(\sqrt {1 + \dfrac{b^2}{a^2}}\) 
y^{2}/b^{2}  x^{2}/a^{2} = 1  (0, ± be), where e = \( \sqrt {1 + \dfrac{a^2}{b^2}}\) 
(x  h)^{2}/a^{2}  (y  k)^{2}/b^{2} = 1  (h ± c, k), where c^{2} = a^{2} + b^{2} 
(y  k)^{2}/b^{2}  (x  h)^{2}/a^{2} = 1  (h, k ± c), where c^{2} = a^{2} + b^{2} 
Properties of Foci of Hyperbola
The following properties of the foci of hyperbola help in a better understanding of the foci of hyperbola.
 There are two foci for the hyperbola.
 The foci lie on the axis of the hyperbola.
 The foci of the hyperbola is equidistant from the center of the hyperbola.
 The foci of hyperbola and the vertex of hyperbola are collinear.
Terms Related to Foci of Hyperbola
The following concepts help in an easier understanding of the foci of the hyperbola.
 Vertex of Hyperbola: The vertex of hyperbola is a point on the axis, where the hyperbola cuts the axis. For the hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\), the two vertices are (+a, 0), and (a, 0). The two vertices 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 Foci of Hyperbola

Example 1: Find the foci of the hyperbola having the vertices at the points (+5, 0), and an eccentricity of 3/2.
Solution:
The given vertex of hyperbola is (+a, 0) = (+5, 0).
Eccentricity of hyperbola = e = 3/2
Foci of hyperbola = (+ae, 0) = (+5 × 3/2, 0)= (+7.5, 0)
Answer: Therefore the two foci of hyperbola are (+7.5, 0), and (7.5, 0).

Example 2: Find the foci of hyperbola having the the equation \(\dfrac{x^2}{36}  \dfrac{y^2}{25}=1\).
Solution:
The given equation of hyperbola is \(\dfrac{x^2}{36}  \dfrac{y^2}{25}=1\)
Comparing this with the standard equation of Hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2}=1\), we have \(a^2 = 36\), and \(b^2 = 25\).
a = 6, and b = 5. Let us now find the eccentricity 'e' of the hyperbola.
\(e = \sqrt {1 + \dfrac{b^2}{a^2}}\)
\(e = \sqrt {1 + \dfrac{6^2}{5^2}}\)
\(e = \sqrt {\dfrac{25 + 36}{25}}\)
\(e = \sqrt {\dfrac{61}{25}}\)
\(e = \dfrac{7.8}{5}\)
e = 1.56
The required foci of hyperbola is (+ae, 0) = (+6 × 1.56, 0) = (+9.37, 0)
Answer: Therefore the foci of the hyperbola are (+9.37, 0), and (9.37, 0).
FAQs on Foci of Hyperbola
What Is Foci Of Hyperbola?
The foci of the hyperbola are the two points on the axis of the hyperbola. The two foci of the hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) are (+ae, 0), and (ae, 0).
How Many Foci Does A Hyperbola Have?
The hyperbola has two foci. For the hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) the two foci are (+ae, 0), and (ae, 0).
How To Find Foci Of Hyperbola From Equation Of Hyperbola?
The foci can be computed from the equation of hyperbola in two simple steps. From the equation of hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\), the value of 'a' can be obtained. The eccentricity of hyperbola can be computed using the formula \(e = \sqrt {1 + \dfrac{b^2}{a^2}} \). Thus the two foci of the hyperbola can be computed from the coordinates (+ae, 0), and (ae, 0).
What Is The Difference Between Foci And Focus Of Hyperbola?
The foci and focus of hyperbola refer to the same. The foci is the plural of focus. Since the hyperbola has two focus, it is referred as foci of hyperbola.
What Is The Use Of Foci Of Hyperbola?
The foci of hyperbola is helpful to find the eccentricity of the hyperbola, and also is useful to further find the equation of hyperbola.
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