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General info on standard forms - Circle, Hyperbole, etc.

Posted on 2004-08-20
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Last Modified: 2008-02-26
This question is intended as an extension of a previous question:

http://www.experts-exchange.com/Miscellaneous/Math_Science/Q_21081042.html

And, because of generous contributions made by PointyEars in the above link, I would like to designate 100 points to PointyEars (provided some comment is made here).

I am looking for general information regarding standard forms,

For example:
How many "standard forms" are there?

For the parabola:
(x-h)² = 4p(y-k)   - http://www.gomath.com/Questions/question.php?question=13281
y=a(x-p)²+q        - From my text.
y = A*x²             - From snoyes_jw
y = A*x² + B

I want to be able to recognize variations of standard forms. I also want to make it interesting. The comments made by PointyEars in the link above did just that. Hoping for more of the same.

Thanks
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Question by:CodeFish
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by:ozo
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y=A*x²+B*x+C
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PointyEars earned 180 total points
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Let's see...   I am scraping the bottom of the pot here  ;-)

This comment is a detailed analysis of some (not all) quadratic equations.  I cannot invest more time in it, but I hope that you will find it helpful...

All the curves we have been talking about are polynomial.  That is, of the type:
  sum( A_kn * x^k * y^(n-k) ) = 0      sum for all k = 0 .. n  and for all n = 0 .. N
where A_kn are numeric coefficients (i.e. constants) dependent on k an n, and N is an integer called the degree of the equation.  This is the most generic form of such equation you can write.

Each term has a degree given by the product of the exponents of x and y.  Obviously, some of the coefficients can be zero (i.e. some of the terms can be missing), but if all the terms of degree N are missing, the equation "degenerates" into an equation of degree N-1.

The equations of degree 0 are no equations at all.  They degenerate into the identity 0 = 0.

All possible equations of degree 1 are:
  a*x = 0
  a*x + c = 0
  b*y = 0
  b*y + c = 0
  a*x + b*y = 0
  a*x + b*y + c = 0
When plotted on the xy axes, these equations are all straight lines.  The most generic is the last one.  The first one is the equation of the y-axis, the second one of a parallel to the y-axis, ..., the second last is a line passing through the origin at an angle.

We have been talking of equation with N = 2 (i.e. quadratic).  The full list of possibilities for N=2 (not degenerate) consists of equations that we can group as follows:

=== only x
  a*x² = 0
  a*x² + b*x = 0
  a*x² + c = 0
  a*x² + b*x + c = 0
These equations are solvable in x and can be written as (x - x1)*(x - x2) = 0, with x1 possibly = x2.  This means that, when plotted, these equations produce two (or just one) straight lines parallel to the y-axis: x = x1 and x = x2.

=== only y:
Like case 1 but with line[s] parallel to the x-axis

=== both x and y:
The most generic equation in this group is:
  a*x² + b*y² + c*x*y + d*x + e*y + f = 0
There are 6 terms in total.  All possible equations with 2 terms :
  a*x² + b*y² = 0    =>  sqrt(a)*x = ± sqrt(-b)*y  =>  two lines passing through the origin
  a*x² + c*x*y = 0   =>  x * (a*x + c*y) = 0  => y-axis and one line through the origin
  a*x² + e*y = 0    => parabola with axis on the y-axis and apex in the origin
  b*y² + c*x*y = 0  =>  y * (c*x + b*y) = 0  => x-axis and one line through the origin
  b*y² + d*x = 0  => parabola with axis on the x-axis and apex in the origin
All possibile equations with 3 terms.  These are quite a few...

  a*x² + b*y² + c*x*y = 0
    can be written as
      y² * [a*(x/y)² + b + c*x/y] = 0
    The expression in square brackets can be solved in x/y:
      y²*(x/y - z1)*(x/y - z2) = 0
    with z1,z2 constants  =>
      (x - z1*y)*(x - z2*y) = 0   =>  two straight lines
  a*x² + b*y² + d*x = 0
    if you replace x with t - d/(2a) you get
    a*t² + b*y² = d²/(4a)  =>  ellipse centred in (d/2a, 0)  (circle if a = b)
  a*x² + b*y² + e*y = 0  =>  ellipse centred in (0, e/2b)  (circle if a = b)
  a*x² + b*y² + f = 0  =>  ellipse (circle if a = b)

  a*x² + c*x*y + d*x = 0
    can be written as
      x * (a*x + c*y + d) = 0  =>  two lines
  a*x² + c*x*y + e*y = 0   =>  ZZZ
  a*x² + c*x*y + f = 0  =>  ZZZ

  a*x² + e*y + d*x  = 0    =>  a parabola with axis parallelto the y-axis
  a*x² + e*y + f  = 0    =>  ditto

  b*y² + c*x*y + d*x  = 0  =>  ZZZ
  b*y² + c*x*y + e*y = 0  =>  two lines
  b*y² + c*x*y + f  = 0  =>  ZZZ

  b*y² + d*x + e*y = 0  =>  a parabola with axis parallelto the x-axis
  b*y² + d*x + f = 0  =>  ditto

For the equations marked with ZZZ I don't immediately identify the curve.  Perhaps I will work on them later (but don't hold your breath!   :-)

With 4 terms, the analysis becomes more difficult. All possible equations with 4 terms containing both x and y are:
  a*x² + b*y² + c*x*y + d*x  = 0
  a*x² + b*y² + c*x*y + e*y  = 0
  a*x² + b*y² + c*x*y + f  = 0
  a*x² + b*y² + d*x + e*y = 0
  a*x² + b*y² + d*x + f = 0
  a*x² + b*y² + e*y + f = 0
  a*x² + c*x*y + d*x + e*y  = 0
  a*x² + c*x*y + d*x + f  = 0
  a*x² + c*x*y + e*y + f = 0
  a*x² + e*y + d*x + f  = 0
  b*y² + c*x*y + d*x + e*y  = 0
  b*y² + c*x*y + d*x + f  = 0
  b*y² + c*x*y + e*y + f  = 0
  b*y² + d*x + e*y + f = 0

There are only 6 choices of 5-term equations, each obtained by leaving out one of the 6 terms.
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by:ozo
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A general connic section is
A*x² + B*y² + C*x*y + D*x  + E*y + F= 0
if we evaluate
| A    C/2    D/2|
|C/2    B     E/2|  =H
|D/2   E/2     F |

| A  C/2|
|C/2  B |  = J
then
when H != 0 and J < 0 we have a hyperbola
when H != 0 and J = 0 we have a parabola
when H != 0 and J > 0 we have an ellipse
When H = 0 and J < 0 we have intersecting lines
when H = 0 and J > 0 we have a point
when H = 0 and J = 0 we have parallel lines
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