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Hyperbola is one of the types of conic section. It is similar to an ellipse. The only difference between the hyperbola and ellipse is that, in the ellipse the sum of the distances between the foci and the point on the ellipse is fixed. Whereas, in a hyperbola the difference between the foci and point on hyperbola is fixed. Therefore, a hyperbola is the set of all points (x, y) in a plane, where the differences between the foci and a point on hyperbola is fixed.

A hyperbola can also be defined as the locus of all points of the plane whose distances to two fixed points is constant.

Hyperbola is a smooth curve and an unbounded case of the conic section, formed by the intersection of a plane with both halves of a double cone. The shape of a Hyperbola is well-defined by its eccentricity e, which is a dimensionless number and always greater than one (e > 1).

Equation of Hyperbola

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The derivation of hyperbola equation is similar to that of an ellipse. The equation of hyperbola is derived from its definition. The below figure shows the orientation of both the horizontal and vertical transverse axis of a hyperbola.

When the transverse axis is horizontal:


Equation of hyperbola, when the transverse axis is horizontal with the center (h, k) is given as follows:

Equation of Hyperbola
Here, a is the units of the vertices from the center and c is the units of the foci from the center. We know that $c^2 = a^2 + b^2$.

And, at the origin (0, 0), the equation of hyperbola is given as follows:
Hyperbola Equation
When the transverse axis is vertical:


Equation of the hyperbola, when the transverse axis is vertical with center (h, k) is given as follows:

Hyperbola Equations
And, when the origin is (0, 0), then the equation of hyperbola is written as follows:

Equations of Hyperbola

Graph of Hyperbola

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The graph of a hyperbola consist of two parallel branches. The line which goes crossing the foci on the branches of hyperbola are called as vertex. There are 2 vertices in a hyperbola. The line segment connecting the vertices is the transverse axis. The midpoint of transverse axis is called as center of the hyperbola.

The values of a and c keeps varying from one hyperbola to another, but they are always fixed to the given values for any hyperbola.

Graph of Hyperbola

As mentioned earlier, for any point on the hyperbola, the difference of the two foci is fixed. In the above simple graph of the hyperbola, you see a point on one of the vertices. This fixed distance must be less. That is, (a + c) - (c - a) = 2a.

Eccentricity of Hyperbola

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The eccentricity is the parameter that is present with all the conic section. It is a degree to measure how much the conic section deviates from being circular. The eccentricity of a hyperbola is greater than 1 and is denoted by e.

Asymptotes of Hyperbola

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Asymptotes are the one which intersect at the center of the hyperbola and there are two asymptotes. The asymptotes pass through the vertices of a rectangle dimension 2a by 2b, with its center at (h, k). The line segment of length 2b joining (h, k + b) and (h, k - b) is the conjugate axis of the hyperbola.

Asymptotes of Hyperbola

The asymptotes of a hyperbola are given as follows:

Horizontal Transverse Axis

$y$ = $k \pm$$ \frac{b}{a}$$(x - h)$

Vertical Transverse Axis

$y$ = $k \pm$$ \frac{a}{b}$$(x - h)$

Directrix of Hyperbola

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The directrix of the hyperbola is line which together with the point is called as focus. The directrix is constant e. Therefore, it is larger than 1. The constant e is the eccentricity of the hyperbola. Hyperbola has two directrices which are parallel to the conjugates axis and tangent to the hyperbola at a vertex.

Directrix of Hyperbola

The equation of directrix is given as follows:

$x$ = $\frac{a^2}{c}$

Foci of Hyperbola

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Foci are the two fixed points, lying inside each curve of the hyperbola. The hyperbola is defined on the basis of foci.

Foci of Hyperbola

The hyperbola with foci is given as $d_1 - d_2$, which is the constant on the hyperbola. If (x, y) is on the other part of hyperbola, then $d_2 - d_1$ is equal to the same constant.

A foci on a horizontal hyperbola is given as follows:

Hyperbola Foci
And, a foci on a vertical hyperbola is given as follows:

Hyperbola Focus

Examples on Hyperbola

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Given below are few problems based on hyperbola:

Solved Example

Question: Find the eccentricity of the hyperbola 9x2 - 4y2 = 36.
Given: 9x- 4y2 = 36

By dividing the whole equation by 36, we get

$\frac{x^2}{4}$ - $\frac{y^2}{9}$ = 1

$\therefore$ a= 4, and b= 9

The eccentricity, e is given by $e^2$ = 1 + $\frac{b^2}{a^2}$ = 1 + $\frac{9}{4}$

$e^2$ = $\frac{13}{4}$

$\therefore$ e = $\frac{\sqrt13}{2}$