Finding the Asymptotes of a Hyperbola

[tade]

Given a hyperbola of the form:

$$r=\frac{A}{1+Bsinθ+Dcosθ}$$

what are the polar equations for the asymptotes?

 

[tade]

Oddly, plotting this with a computer will result in the asymptotes being plotted as well.

 

[TitoSmooth]

You can always change it into cartesian coordinates but the work can be tedious. I have not done conics in polar plane in a long time, but I can add snapshots of of a book on how to do so If you would like.

 

[Office_Shredder]

An asymptote is going to necessarily have r going to infinity, so the denominator has to be going to zero.

 

[CellCoree]

The polar equations for the asymptotes of a hyperbola in the form given above can be derived by setting the denominator of the equation equal to zero, since the asymptotes occur when the hyperbola approaches infinity. This results in the following equations:

θ = arctan(-B/D) + nπ

where n is any integer.

These equations represent the angles at which the asymptotes intersect the origin. To find the polar equations for the asymptotes themselves, we can substitute these angles back into the original equation for the hyperbola. This results in the following equations:

r = A/(1 + (B/D)sinθ + (B/D)cosθ)

and

r = -A/(1 + (B/D)sinθ + (B/D)cosθ)

These equations represent the two asymptotes of the hyperbola, with the first equation corresponding to the asymptote at θ = arctan(-B/D) + nπ and the second equation corresponding to the asymptote at θ = arctan(-B/D) + (n+1)π.

It should also be noted that the values of A, B, and D in the original hyperbola equation determine the shape and orientation of the hyperbola, and therefore, the values of the asymptotes. As such, the polar equations for the asymptotes will also vary depending on the specific values of these parameters.