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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

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