HYPERBOLA FORMULA

In simple sense, hyperbola looks similar to to mirrored parabolas. The two halves are called the branches. When the plane intersect on the halves  of a right circular cone angle of which will be parallel to the axis of the cone, a parabola is formed.

A hyperbola contains: two foci and two vertices. The foci of the hyperbola are away from the hyperbola’s center and vertices. Here is an illustration to make you understand:

Hyperbola Formula

The equation for hyperbola is,

\[\large \frac{(x-x_{0})^{2}}{a^{2}}-\frac{(y-y_{0})^{2}}{b^{2}}=1\]

Where,
$x_{0}, y_{0}$ are the center points.
$a$ = semi-major axis.
$b$ = semi-minor axis.

Let us learn the basic terminologies related to hyperbola formula:

MAJOR AXIS

The line that passes through the center, focus of the hyperbola and vertices is the Major Axis. Length of the major axis = 2a. The equation is given as:

\[\large y=y_{0}\]

MINOR AXIS

The line perpendicular to the major axis and passes by the middle of the hyperbola is the Minor Axis.
Length of the minor axis = 2b. The equation is given as:

\[\large x=x_{0}\]

ECCENTRICITY

The variation in the conic section being completely circular is eccentricity. It is usually greater than 1 for hyperbola. Eccentricity is $2\sqrt{2}$ for a regular hyperbola. The formula for eccentricity is:

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

ASYMPTOTES

Two bisecting lines that is passing by the center of the hyperbola that doesn’t touch the curve is known as the Asymptotes. The equation is given as:

\[\large y=y_{0}+\frac{b}{a}x-\frac{b}{a}x_{0}\]

\[\large y=y_{0}-\frac{b}{a}x+\frac{b}{a}x_{0}\]

Directrix of a hyperbola

Directrix of a hyperbola is a straight line that is used in generating a curve. It can also be defined as the line from which the hyperbola curves away from. This line is perpendicular to the axis of symmetry. The equation of directrix is:

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

VERTEX

The point of the branch that is stretched and is closest to the center is the vertex. The vertex points are

$\LARGE \left(a,y_{0}\right ) and $\LARGE \left(-a,y_{0}\right )$

Focus (foci)

On a hyperbola, focus (foci being plural) are the fixed points such that the difference between the distances are always found to be constant. The two focal points are: 

\[\large\left(x_{0}+\sqrt{a^{2}+b^{2}},y_{0}\right)\]

\[\large \left(x_{0}-\sqrt{a^{2}+b^{2}},y_{0}\right)\]3

Solved examples

Question: The equation of the hyperbola is given as: $\frac{(x-4)^{2}}{9^{2}}-\frac{(y-2)^{2}}{7^{2}}$ 

Find the following: Vertex, Asymptote, Major Axis, Minor Axis and Directrix?

Solution:

Given,
$x_{0}=4$
$y_{0}=2$
$a =9$
$b = 7$

The vertex point: $(a, y_{0})$ and $(-a, y_{0})$ are: $(9, 2)$ and $(-9, 2)$

Asymptote

$y=2+\frac{7}{9}x-\frac{37}{9}$

$y=2-\frac{7}{9}x-\frac{23}{9}$

$y=2+0.77x+4.1=6.1+0.77x$

$y=2-0.77x+2.5=4.5+0.77x$

Major Axis

$y=y_{o}$

$y_{o}=2$

Minor Axis

$x=x_{o}$

$x_{o} =4$

Directrix

$x=\frac{\pm 9^{2}}{\sqrt{9^{2}+7^{2}}} = \pm \frac{81}{\sqrt{81+49}}=7.1$


Practise This Question

A charge is fixed at one end of an insulating rod of length ‘L’ which rotates with uniform speed about an axis perpendicular to its length and passing through its other end as shown in the figure. This region of space contains a uniform electrostatic field E directed along positive X-axis. The work done by the field when the particle completes two full rotations is?