Asymptotes of a conic section

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 :

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

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:

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


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