Foci Of Hyperbola - Definition, Formula, Properties, FAQs - Cuemath

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.

1.	What Is Foci of Hyperbola?
2.	Foci of Hyperbola Formulas
3.	Properties of Foci of Hyperbola
4.	Terms Related to Foci of Hyperbola
5.	Examples on Foci of Hyperbola
6.	Practice Questions
7.	FAQs on Foci of Hyperbola

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
x2/a2 - y2/b2 = 1	(± ae, 0), where e = \(\sqrt {1 + \dfrac{b^2}{a^2}}\)
y2/b2 - x2/a2 = 1	(0, ± be), where e = \( \sqrt {1 + \dfrac{a^2}{b^2}}\)
(x - h)2/a2 - (y - k)2/b2 = 1	(h ± c, k), where c2 = a2 + b2
(y - k)2/b2 - (x - h)2/a2 = 1	(h, k ± c), where c2 = a2 + b2

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 2b2/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 x-axis 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 \(\dfrac{x^2}{a^2} - \dfrac{y^2}{a^2} = 1\).

☛Related Topics

• Rectangular Hyperbola
• Latus Rectum of Ellipse
• Vertex of Ellipse
• Area of Ellipse
• Axis of Symmetry of Parabola