Hyperbola Formula

Hyperbola is the part of the conic section. It is a plane geometry which is generated by a point which moves in plane such that the difference of the distance between two fixed focuses is always constant. 

Terms of Hyperbola

Major axis : The major axis of the hyperbola is a line that passes through the foci, center and vertices of the hyperbola. 

Minor axis : The minor axis of hyperbola is the perpendicular line to major axis and pasess through the center of the hyperbola.

Vertices : Vertices are the point on the axis of the hyperbola where hyperbola passes the axis.

Foci : The hyperbola has two focus and both are equal distances from the center of the hyperbola and it is collinear with vertices of the hyperbola

Equation of Hyperbola 

The hyperbola equation is,

$\dfrac{({x-x_0})^2}{a^2}-\frac{({y-y_0})^2}{b^2}=1$

Where,

$x_0$, $y_0$ = The center points.

a = Semi-major axis.

b = Semi-minor axis.

All Formula of Hyperbola

Let’s refer to the hyperbola formula table and learn the basic terminologies with respect to the hyperbola formula:

1. Minor Axis

The line perpendicular to the major axis and crosses through the centre of the hyperbola is the Minor Axis.

The length of the minor axis is 2b. The equation is as follows:

x = $x_0$

2. Major Axis

The line that crosses by the middle, the focus of the hyperbola and vertices is the Major Axis. The length of the major axis is 2a. The equation is as follows:

y = $y_0$

3. Eccentricity

The differentiation in the conic section being fully circular is eccentricity. It is generally higher than 1 for hyperbola. Eccentricity is 2√2 for a regular hyperbola. The eccentricity formula is: 

$\dfrac{\sqrt{a^{2}+b^{2}}}{a}$

4. Asymptotes

Two intersecting line segments that are crossing through the centre of the hyperbola which does not touch the curve are called the Asymptotes. The asymptotes formula is given as:

y = $y_0 + \dfrac{bx}{a} – \dfrac{bx}{x_0}$

y = $y_0 - \dfrac{bx}{a} + \dfrac{bx}{x_0}$

5. Vertices : Vertices are the point on the axis of the hyperbola where hyperbola passes the axis. The coordinates of vertices are (a,$y_0$) and (-a,$y_0$).

Hyperbola Foci Formula 

The line segment across the foci is known as the transverse axis. Also, the line segment through the midpoint and perpendicular to the transverse axis is known as the conjugate axis. The points at which the hyperbola bisects the transverse axis are referred to as the vertices of the hyperbola.

• The distance between the two foci is: 2c

• The length of the conjugate axis is 2b… in which b = $\sqrt{c^2-a^2}$

• The distance between two vertices is: 2a (i.e. also the length of the transverse axis)

Directrix of Hyperbola

The directrix of a hyperbola is a straight line that is used in incorporating a curve. It can also be described as the line segment from which the hyperbola curves away. This line segment is perpendicular to the axis of symmetry. The equation of directrix formula is as follows:

x = \[\frac{ a^{2}}{\sqrt{a^{2}+ b^{2}}}\]

Solved Example

Example 1. The equation of hyperbola is given below:

$\dfrac{({x-2})^2}{6^2}-\frac{({y+5})^2}{3^2}=1$

Find the vertices and length of minor axis and major axis.

Sol: 

Given equation of hyperbola $\dfrac{({x-2})^2}{6^2}-\frac{({y+5})^2}{3^2}=1$.

Compare the given equation with general equation of hyperbola

$\dfrac{({x-x_0})^2}{a^2}-\frac{({y-y_0})^2}{b^2}=1$.

Coordinate Vertices = (a,$y_0$) and (-a,$y_0$)

Length of minor axis = 2b

Length of major axis = 2a

Value of $y_0$= -5

And a = 6 , b = 3

Then Coordinate Vertices of given hyperbola = (6,-5) and (-6,-5).

Length of minor axis = 6.

Length of major axis = 12.

Conclusion

Hyperbola is an important part of the conic section. They are important topics to learn as students will remember and carry them with them for higher classes. Allowing students to visualise and conceptualise Different terms of hyperbola and their formula  their knowledge of the topics and will help them develop skills not only in this topic, but also in related ones.