HYPERBOLA FORMULA

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,

\(\begin{array}{l}x_{0}, y_{0}\end{array} \)
are the center points.
\(\begin{array}{l}a\end{array} \)
= semi-major axis.
\(\begin{array}{l}b\end{array} \)
= 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

\(\begin{array}{l}2\sqrt{2}\end{array} \)
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

\(\begin{array}{l}\LARGE \left(a,y_{0}\right ) \ and \ \LARGE \left(-a,y_{0}\right )\end{array} \)

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: 

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

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

Solved examples

Question: The equation of the hyperbola is given as: 

\(\begin{array}{l}\frac{(x-4)^{2}}{9^{2}}-\frac{(y-2)^{2}}{7^{2}}\end{array} \)
 

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

Solution:

Given,

\(\begin{array}{l}x_{0}=4\end{array} \)
\(\begin{array}{l}y_{0}=2\end{array} \)
\(\begin{array}{l}a =9\end{array} \)
\(\begin{array}{l}b = 7\end{array} \)

The vertex point: 

\(\begin{array}{l}(a, y_{0})\end{array} \)
and 
\(\begin{array}{l}(-a, y_{0})\end{array} \)
are
\(\begin{array}{l}(9, 2)\end{array} \)
and
\(\begin{array}{l}(-9, 2)\end{array} \)

Asymptote

\(\begin{array}{l}y=\frac{7}{9}(x-4)+2\\ y=-\frac{7}{9}(x-4)+2\end{array} \)

Major Axis

a = 9

Minor Axis

b = 7

Directrix

\(\begin{array}{l}x=\frac{\pm 9^{2}}{\sqrt{9^{2}+7^{2}}} = \pm \frac{81}{\sqrt{81+49}}=7.1\end{array} \)

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