How to Find the Asymptotes of a Hyperbola

Hyperbola

The hyperbola is a conic section. The term hyperbola is referred to the two disconnected curves shown in the figure.how to find asymptotes of a hyperbola | Pediaa.com

If the principal axes are coinciding with the Cartesian axes, the general equation of hyperbola is of the form:

How to find asymptotes of a hyperbola | Pediaa.com

These hyperbolas are symmetric around y axis and are known as y-axis hyperbola. The hyperbola symmetric around x-axis (or x-axis hyperbola) are given by the equation,

How to find asymptotes of a hyperbola | Pediaa.comHow to find the asymptotes of a hyperbola 

To find the asymptotes of a hyperbola, use a simple manipulation of the equation of the parabola.

i. First bring the equation of the parabola to above given form

If the parabola is given as mx2+ny2=l, by defining  

a=√(l/m) and b=√(-l/n) where l<0

(This step is not necessary if the equation is given in standard from.

How to find asymptotes of a hyperbola | Pediaa.com

ii. Then, replace the right hand side of the equation with zero.

How to find asymptotes of a hyperbola | Pediaa.com

iii. Factorize the equation and take solutions

How to find asymptotes of a hyperbola | Pediaa.com

Therefore, the solutions are ,

How to find asymptotes of a hyperbola | Pediaa.com

Equations of the asymptotes are

How to find asymptotes of a hyperbola | Pediaa.com

Equations of the asymptotes for the x-axis hyperbola also can be obtained by the same procedure.

Find the asymptotes of a hyperbola – Example 1

Consider the hyperbola given by the equation x2/4-y2/9=1. Find the equations of the asymptotes.

How to find asymptotes of a hyperbola | Pediaa.com

Rewrite the equation and follow the above procedure.
x2/4-y2/9=x2/22 -y2/32 =1

By replacing the right hand side with zero, the equation becomes x2/22 -y2/32 =0.
Factorizing and taking solution of the equation give,

(x/2-y/3)(x/2+y/3)=0

Equations of the asymptotes are,

3x-2y=0 and 3x+2y=0

Find the asymptotes of a hyperbola – Example 2

  • Equation of a parabola is given as -4x² + y² = 4

How to find asymptotes of a hyperbola | Pediaa.com

This hyperbola is an x-axis hyperbola.
Rearranging the terms of the hyperbola into the standard from gives
-4x2+ y2= 4=>y2/22 -x2/12 =1
Factorizing the equation provides the following
(y/2-x)(y/2+x)=0
Therefore, the solutions are y-2x=0 and y+2x=0.

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