Asymptotes of a conic section

• Direction of asymptotes
• Calculating the directions of the asymptotes
• Remarks :
• Asymptote of a conic section
• Remark :
• Calculating the asymptotes
• General method
• Special cases:
• Special method to calculate the asymptotes
• Theorem
• Quadratic equation of the asymptotes of an ellipse or hyperbola.
• Equation of a conic section with given asymptotes

Direction of asymptotes

The directions of the asymptotes are the directions defined by the ideal points of the conic section.

Calculating the directions of the asymptotes

Example:

 
 
 F(x,y,z) = 3 x2 - y2 + 2 xy + 4 x - 2 y + 7 = 0
 
 Asymptote has slope m
 
<=>
 (1,m,0) is on the conic section
 
<=>
 3 + 2 m - m2 = 0
<=>
 m = -1 or m = 3

Remarks :

• A parabola has two equal ideal points. It is THE asymptotic direction of the parabola.
• If a' = 0, the direction (0,1,0) is an asymptotic direction.
• The asymptotic directions do not depend on a".

Asymptote of a conic section

An asymptote of a conic section is the tangent line in an ideal point of the conic section.

Remark :

A parabola has two equal ideal points. If the parabola is not degenerated, that ideal point is a simple point and the tangent line is the ideal line. So, the ideal line is the asymptote of a not degenerated parabola.

Calculating the asymptotes

General method

• Calculate the ideal points
• Calculate the tangent lines

Example:

 
 F(x,y,z) = 3 x2 - y2 + 2 xy + 4 x - 2 y + 7 = 0

The ideal points are (1,-1,0) and (1,3,0).

The tangent line in (1,-1,0) is 3 x + y + 3 = 0

The tangent line in (1, 3 ,0) is 3 x - 3 y - 1 = 0

Special cases:

• The ideal line is the asymptote of a non-degenerated parabola.
• A degenerated parabola has a double point as ideal point. thus, each line through that point is an asymptote.
• The asymptotes of a degenerated ellipse or hyperbola are coinciding with the components.

Special method to calculate the asymptotes

Say (1,m,0) is an ideal point of the conic section. The value m does not depend on a". The asymptote has equation

 
 Fx' (x,y,z) + m Fy' (x,y,z) = 0
<=>
 ( a x + b" y + b' z ) + m ( b" x + a' y + b z ) = 0

So, we see that the asymtote does not depend on a" . This property can be useful to calculate the asymptotes.
Example:

 
 x2 - xy - 2 x - 5 = 0
 
has the same asymptotes as
 x2 - xy - 2 x = 0
<=>
 x (x - y - 2) = 0
 
The asymptotes are x = 0 and x - y - 2 = 0

Theorem

It can be proved that :
If two ellipsis or two hyperbolas have the same asymptotes, then their equations can be written such that only the a" differs.

Quadratic equation of the asymptotes of an ellipse or hyperbola.

Example: Take the conic section

 
 x2 - x y - 2 y2 + 3 x + 3 y + 7 = 0
 
It has the same asymptotes as
 
 x 2 - x y - 2 y2 + 3 x + 3 y + k = 0
 
Now, choose k such that the conic section is degenerated.
The condition is
 
 DELTA = 0
<=>
 | 2, -1, 3 |
 | -1, -4, 3 | = 0
 | 3, 3, 2 k |
<=>
 -18 k = 0
<=>
 k = 0
 
Therefore, the quadratic equation of the asymptotes is
 
 x2 - x y - 2 y2 + 3 x + 3 y = 0

Equation of a conic section with given asymptotes

 
Say u1 x + v1 y + w1 = 0
 u2 x + v2 y + w2 = 0

are the asymptotes of a conic section.

The degenerated conic section with these asymptotes is

 
 (u1 x + v1 y + w1)(u2 x + v2 y + w2) = 0

All conic sections with these asymptotes have equation

 
 (u1 x + v1 y + w1)(u2 x + v2 y + w2) + h = 0