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Hyperbola Calculator
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A Hyperbola is a curve on the plane made of two branches. It is formed when the plane intersects the halves of a right circular cone whose angle is parallel to the minor and major axis of the cone.It is given by the equation:

$\frac{x^{2} - x_{0}^{2}}{a^{2}}$ + $\frac{y^{2} - y_{0}^{2} }{b^{2}}$ = 1.
where C (xo,yo) are the center points and a and b are semi-major and semi-minor axis respectively.

Hyperbola Calculator calculates the focii, eccentricity and asymptotes if the center coordinate points and the value of a and b are given.

## Steps for Hyperbola Calculator

Step 1 :

Read the problem and observe whether it is of the form
$\frac{(x - h)^{2}}{a^{2}}$ + $\frac{(y - k)^{2}}{b^{2}}$ = 1.
where (h,k) are centre points of hyperbola.

Step 2 :

To find the terminologies related to hyperbola use the below formula:
Hyperbola Focus F X Coordinate  =  x0 + $\sqrt{(a^{2} + b^{2})}$.
Hyperbola Focus F Y Coordinate  =  y0
To find Focii image use below formula:
Hyperbola Focus F' X Coordinate  =  x0 - $\sqrt{(a^{2} + b^{2})}$
Hyperbola Focus F' Y Coordinate  =  y0
The Asymptotes are given by
Asymptotes H'L: y = $\frac{b}{a}$ x + y0 - $\frac{b}{a}$ x0
Asymptotes LH': y = - $\frac{b}{a}$ x + y0 + $\frac{b}{a}$ x0
The eccentricity of the hyperbola is given by e =  $\frac{\sqrt{(a^{2} + b^{2})}}{a}$.

Substitute the values in these to get the desired parameter.

## Problems on Hyperbola Calculator

1. ### Find the focii and asymptotes of a Hyperbola which is diven by the equation:$\frac{(x - 2)^{2}}{16}$ + $\frac{(y - 3)^{2}}{81}$ = 1.

Step 1 :

given equation is $\frac{(x - 2)^{2}}{16}$ + $\frac{(y - 3)^{2}}{81}$ = 1,

Here centre points are (2,3), a = 4 and b = 9.

Step 2 :

Hyperbola Focus FX Coordinate  =  x0 + $\sqrt{a^{2} + b^{2}}$.

=  2 + $\sqrt{4^{2} + 9^{2}}$

=  2 + $\sqrt{97}$

=  11.848.

Hyperbola Focus FY coordinate = y0 = 3

To find Focii image use below formula:
Hyperbola Focus F' X Coordinate  =  x0 - $\sqrt{(a^{2} + b^{2})}$

=  2 - $\sqrt{4^{2} + 9^{2}}$

=  - 7.848
Hyperbola Focus F' Y Coordinate  =  y0 = 3
The Asymptotes are given by

Asymptotes H'L: y = - $\frac{b}{a}$ x + y0 + $\frac{b}{a}$ x0

= - $\frac{9}{4}$ x + 3 + $\frac{9}{4}$ $\times$ 2

= - 2.25 x + 7.5.

Asymptotes L'H: y = $\frac{9}{4}$ x + 3 - $\frac{9}{4}$$\times$ 2

= $\frac{9}{4}$ x + 3 - 4.5

=  2.25 x - 1.5

Hyperbola Focus F X Coordinate  =  11.848
Hyperbola Focus F Y Coordinate  =  3
Focii mirror image is given as:
Hyperbola Focus F' X Coordinate  =  -7.848
Hyperbola Focus F' Y Coordinate = 3
The Asymptotes are given by
Asymptotes H'L = - 2.25 x + 7.5 and L'H = 2.25 x - 1.5.

2. ### Find the eccentricity of the hyperbola given by the equation:$\frac{(x - 2)^{2}}{16}$ + $\frac{(y - 3)^{2}}{81}$ = 1

Step 1 :

given equation is $\frac{(x - 2)^{2}}{16}$ + $\frac{(y - 3)^{2}}{81}$ = 1,

Here centre points are (2,3), a = 4 and b = 9.

Step 2 :

The eccentricity of the hyperbola is given by e =  $\frac{\sqrt{(a^{2} + b^{2})}}{a}$

=  $\frac{\sqrt{(4^{2} + 9^{2})}}{4}$

=  $\frac{\sqrt{97}}{4}$

=  2.462.