When the principal axes coincide with the Cartesian axes, the general hyperbolic equation is of the form:
These hyperbolas are symmetrical about the y-axis and are referred to as y-axis hyperbolas. The hyperbola symmetrical about the x-axis (or x-axis hyperbola) is given by the equation:
To find the asymptotes of a hyperbola, use a simple manipulation of the parabolic equation.
I. First, bring the equation of the parabola into the above form
If the parabola is given as mx 2 + ny 2 = l , one defines
a = √ ( l / m ) and b = √ ( -l / n ) where l <0
(This step is not necessary if the equation is given in standard form.
ii. Then replace the right side of the equation with zero.
iii. Factor the equation and take solutions
Hence the solutions
Are equations of the asymptotes
Equations of the asymptotes for the x-axis hyperbola can also be obtained by the same method.
Find the asymptotes of a hyperbola - Example 1
Consider the hyperbola represented by the equation x 2/4-y 2/9 = 1 is given. Find the equations of the asymptotes.
Rewrite the equation and follow the above procedure. x 2/4-y 2/9 = x 2/2 2 -y 2/3 2 = 1
By replacing the right hand side with zero, the equation becomes x 2/2 2 -y 2/3 2 = 0. Factoring and solving the equation give
(x / 2-y / 3) (x / 2 + y / 3) = 0
Are equations of the asymptotes,
3x-2y = 0 and 3x + 2y = 0
Find the asymptotes of a hyperbola - example 2
- The equation of a parabola is given as -4x² + y² = 4
This hyperbola is an x-axis hyperbola. The rearranging the terms y of the hyperbola in the standard of result-4x 2 + y 2 = 4 => 2/2 2 -x 2/1 2 = 1 The factorization of the equation yields the following (y / 2-x) (y / 2 + x) = 0 Hence the solutions y-2x = 0 and y + 2x = 0.