Systems of conic sections




A system of conic sections

Say F1(x,y,z) = 0 and F2(x,y,z) = 0 are the equations of two conic sections with no common component.
All conic sections with equation
 
         l F1(x,y,z) + m F2(x,y,z) = 0
is called a system of conic sections. The real numbers l and m are homogeneous parameters ( not both = 0 ). All conic sections of the system different from F2(x,y,z) = 0, can be written as
 
           F1(x,y,z) + h F2(x,y,z) = 0
with h = a real non-homogeneous parameter.

Theorem 1

If F1(x,y,z) = 0 and F2(x,y,z) = 0 are the equations of two non-degenerated conic sections, then there is always a real value h such that the conic section F1(x,y,z) + h F2(x,y,z) = 0 is degenerated.

Proof:

 
For     F1(x,y,z) + h F2(x,y,z)  = 0


        | a1 + ha2     b1"+ hb2"      b1'+ hb2'|
DELTA = | b1"+ hb2"    a1'+ ha2'      b1 + hb2 |
        | b1'+ hb2'    b1 + hb2       a1"+ ha2"|

<=>

        | ha2 + a1     hb2"+ b1"      hb2'+ b1'|
DELTA = | hb2"+ b1"    ha2'+ a1'      hb2 + b1 |
        | hb2'+ b1'    hb2 + b1       ha2"+ a1"|

<=>

        | ha2     hb2"+ b1"      hb2'+ b1'|  |a1     hb2"+ b1"      hb2'+ b1'|
DELTA = | hb2"    ha2'+ a1'      hb2 + b1 |+ |b1"    ha2'+ a1'      hb2 + b1 |
        | hb2'    hb2 + b1       ha2"+ a1"|  |b1'    hb2 + b1       ha2"+ a1"|

<=>

          |a2     hb2"+ b1"   hb2'+ b1'|  |a1     hb2"+ b1"  hb2'+ b1'|
DELTA = h |b2"    ha2'+ a1'   hb2 + b1 |+ |b1"    ha2'+ a1'  hb2 + b1 |
          |b2'    hb2 + b1    ha2"+ a1"|  |b1'    hb2 + b1   ha2"+ a1"|

<=>

          |a2     hb2"   hb2'+ b1'|  |a1  hb2"  hb2'+ b1'|
DELTA = h |b2"    ha2'   hb2 + b1 |+ |b1" ha2'  hb2 + b1 | +
          |b2'    hb2    ha2"+ a1"|  |b1' hb2   ha2"+ a1"|

          |a2    b1"   hb2'+ b1'|  |a1   b1"  hb2'+ b1'|
        h |b2"   a1'   hb2 + b1 |+ |b1"  a1'  hb2 + b1 |
          |b2'   b1    ha2"+ a1"|  |b1'  b1   ha2"+ a1"|

<=>

             |a2     b2"   hb2'+ b1'|    |a1  b2"  hb2'+ b1'|
DELTA = h2   |b2"    a2'   hb2 + b1 |+ h |b1" a2'  hb2 + b1 | +
             |b2'    b2    ha2"+ a1"|    |b1' b2   ha2"+ a1"|

          |a2    b1"   hb2'+ b1'|  |a1   b1"  hb2'+ b1'|
        h |b2"   a1'   hb2 + b1 |+ |b1"  a1'  hb2 + b1 |
          |b2'   b1    ha2"+ a1"|  |b1'  b1   ha2"+ a1"|

<=>

             |a2     b2"   hb2'|    |a1  b2"  hb2'|
DELTA = h2   |b2"    a2'   hb2 |+ h |b1" a2'  hb2 | +
             |b2'    b2    ha2"|    |b1' b2   ha2"|

          |a2    b1"   hb2'|  |a1   b1"  hb2'|
        h |b2"   a1'   hb2 |+ |b1"  a1'  hb2 | +
          |b2'   b1    ha2"|  |b1'  b1   ha2"|

             |a2     b2"   b1'|    |a1  b2"  b1'|
        h2   |b2"    a2'   b1 |+ h |b1" a2'  b1 | +
             |b2'    b2    a1"|    |b1' b2   a1"|

          |a2    b1"   b1'|  |a1   b1"  b1'|
        h |b2"   a1'   b1 |+ |b1"  a1'  b1 |
          |b2'   b1    a1"|  |b1'  b1   a1"|

<=>

             |a2     b2"   b2'|    |a1  b2"  hb2'|
DELTA = h3   |b2"    a2'   b2 |+ h |b1" a2'  hb2 | +
             |b2'    b2    a2"|    |b1' b2   ha2"|

          |a2    b1"   hb2'|  |a1   b1"  hb2'|
        h |b2"   a1'   hb2 |+ |b1"  a1'  hb2 | +
          |b2'   b1    ha2"|  |b1'  b1   ha2"|

             |a2     b2"   b1'|    |a1  b2"  b1'|
        h2   |b2"    a2'   b1 |+ h |b1" a2'  b1 | +
             |b2'    b2    a1"|    |b1' b2   a1"|

          |a2    b1"   b1'|  |a1   b1"  b1'|
        h |b2"   a1'   b1 |+ |b1"  a1'  b1 |
          |b2'   b1    a1"|  |b1'  b1   a1"|

Since  F2(x,y,z) = 0 is not degenerated,

        |a2     b2"   b2'|
        |b2"    a2'   b2 | is not  0.
        |b2'    b2    a2"|
We have:
        DELTA = 0
<=>
             |a2     b2"   b2'|    |a1  b2"  hb2'|
        h3   |b2"    a2'   b2 |+ h |b1" a2'  hb2 | +
             |b2'    b2    a2"|    |b1' b2   ha2"|

          |a2    b1"   hb2'|  |a1   b1"  hb2'|
        h |b2"   a1'   hb2 |+ |b1"  a1'  hb2 | +
          |b2'   b1    ha2"|  |b1'  b1   ha2"|

             |a2     b2"   b1'|    |a1  b2"  b1'|
        h2   |b2"    a2'   b1 |+ h |b1" a2'  b1 | +
             |b2'    b2    a1"|    |b1' b2   a1"|

          |a2    b1"   b1'|  |a1   b1"  b1'|
        h |b2"   a1'   b1 |+ |b1"  a1'  b1 |    = 0
          |b2'   b1    a1"|  |b1'  b1   a1"|
This equation has degree = 3 and therefore it has always a real root.

Theorem 2

Two conic sections, with no common component, have 4 common points.

Proof:

Remark : From these common points, no three points are collinear, because the two conic sections have no common component .

Basic points and basic conic sections

Take F1(x,y,z) = 0 and F2(x,y,z) = 0 as equations of two conic sections with no common component.
Each conic sections of the system
 
         l F1(x,y,z) + m F2(x,y,z) = 0
goes through the 4 common points of F1(x,y,z) = 0 and F2(x,y,z) = 0. These 4 common points are common points of all the conic sections of the system. These 4 points are called the basic points of the system. The conic sections F1(x,y,z) = 0 and F2(x,y,z) = 0 are called the basic conic sections of the system.

Two arbitrary conic sections of the system go through the four basic points. These two conic sections can be chosen as basic conic sections of the system.

One conic section through one point

Say P is a point different from a basic point of a system of conic sections. Then there is just one conic section of that system that contains point P.

The proof is left as an exercise.

Degenerated conic sections in a system

Theorem:
If there is at least one non-degenerated conic section in a system, then there are at least one and at most three degenerated conic sections in that system.

Proof:
Take a system with basic conic sections F1(x,y,z) = 0 and F2(x,y,z) = 0. Say F2x,y,z) = 0 is not degenerated. An element of the system different from F2 has equation

 
           F1(x,y,z) + h F2(x,y,z) = 0
From above we know that the DELTA of that conic section can be written as a polynomial in h with degree = 3.

Thus, there are at least one and at most 3 real values of h, such that DELTA = 0.

A conic section through 5 points

Take 5 points such that no three points are collinear.
Through 4 of this points there is a system of conics. From this system, there is just one element going through the fifth point.

Systems of circles

If the basic conic sections of a system are circles, all the elements of the system are circles except one.

The proof is left as an exercise.


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