Systems of conic sections

A system of conic sections
Theorem 1
Theorem 2
Basic points and basic conic sections
One conic section through one point
Degenerated conic sections in a system
A conic section through 5 points
Systems of circles

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:

If at least one conic section is degenerated in line d1 and d2, then d1 cuts the other conic section in 2 points, and d2 cuts the other conic section in 2 points.
If both conic sections are not degenerated:
The common points of both conic sections are the solutions of
```
 
        / F₁(x,y,z) = 0
        \ F₂(x,y,z) = 0
<=>
        / F₁(x,y,z) = 0
        \ F₁(x,y,z) + h F₂(x,y,z)  = 0
```
We choose h such that F₁(x,y,z) + h F₂(x,y,z) = 0 is degenerated. The system has 4 solutions and the conic sections have 4 common points.

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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