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uberclay
science forum beginner
Joined: 22 May 2006
Posts: 7
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 Posted: Sun Jun 18, 2006 5:52 am Post subject:
Understanding the general from of a conic as it relates to standard form equations - Grade 11
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I am having a hard time understanding the general form of conics as it
maps to the individual equations in standard form (2D)
Using the following standard forms
(x - h)^2 = 4p(y - k) is a parabola [P]
(x - h)^2 + (y - k)^2 = r^2 is a circle [C]
[(x - h)^2/a^2] + [(y - k)^2/b^2] = 1 or [(x - h)^2/b^2] + [(y -
k)^2/a^2] = 1 is an ellipse [E]
[(x - h)^2/a^2] - [(y - k)^2/b^2] = 1 or [(y - k)^2/a^2] - [(x -
h)^2/b^2] = 1 is a hyperbola [H]
I derive
[P] x^2 - 2hx + h^2 - 4py + 4pk = 0
[C] x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0
[E] a^2 x^2 - a^2 2hx + a^2 h^2 + b^2 y^2 - b^2 2ky + b^2 k^2 - a^2 b^2
= 0
[E] b^2 x^2 - b^2 2hx + b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2 b^2
= 0
[H] a^2 x^2 - a^2 2hx + a^2 h^2 - b^2 y^2 + b^2 2ky - b^2 k^2 - a^2 b^2
= 0
[H] - b^2 x^2 + b^2 2hx - b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2
b^2 = 0
My text states the general form of conics as
Ax^2 + By^2 + Fx + Gy + C = 0
I am guessing that (depending on the type of
conic)
A = a^2 or b^2
B = b^2 or a^2
(A & B = 1 in the cases of circles and parabolas)
F = - A2h
G = - B2k
C = Ah^2 + B k^2 - AB or Ak^2 + B h^2 - AB (with the exception of a
parabola)
Am I correct in the values of A, B, C, F and G?
I have found so many sites that cover the general form but none that
show specifically how the standard and general forms relate.
I have also seen the general form most commonly expressed as
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Would it serve me better to concentrate on this equation? |
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Virgil
science forum Guru
Joined: 24 Mar 2005
Posts: 5536
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| Posted: Sun Jun 18, 2006 6:43 am Post subject:
Re: Understanding the general from of a conic as it relates to standard form equations - Grade 11
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In article <1150609937.105338.286630@c74g2000cwc.googlegroups.com>,
"uberclay" <david.palmatier@gmail.com> wrote:
| Quote: | I am having a hard time understanding the general form of conics as it
maps to the individual equations in standard form (2D)
Using the following standard forms
(x - h)^2 = 4p(y - k) is a parabola [P]
(x - h)^2 + (y - k)^2 = r^2 is a circle [C]
[(x - h)^2/a^2] + [(y - k)^2/b^2] = 1 or [(x - h)^2/b^2] + [(y -
k)^2/a^2] = 1 is an ellipse [E]
[(x - h)^2/a^2] - [(y - k)^2/b^2] = 1 or [(y - k)^2/a^2] - [(x -
h)^2/b^2] = 1 is a hyperbola [H]
I derive
[P] x^2 - 2hx + h^2 - 4py + 4pk = 0
[C] x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0
[E] a^2 x^2 - a^2 2hx + a^2 h^2 + b^2 y^2 - b^2 2ky + b^2 k^2 - a^2 b^2
= 0
[E] b^2 x^2 - b^2 2hx + b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2 b^2
= 0
[H] a^2 x^2 - a^2 2hx + a^2 h^2 - b^2 y^2 + b^2 2ky - b^2 k^2 - a^2 b^2
= 0
[H] - b^2 x^2 + b^2 2hx - b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2
b^2 = 0
My text states the general form of conics as
Ax^2 + By^2 + Fx + Gy + C = 0
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Actually this is only true for conics that have an axis of symmetry
parallel to a coordinate axis.
The more general conic equation is of form
A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0
for suitable constants A, B, C, D, E, AND F.
For equations of form A*x^2 + B*y^2 + F*x + G*y + C = 0,
if A = B <> 0, you have a circle,
if A*B > 0, an ellipse,
if A*B < 0, an hyperbola, and
if A = 0 or B = 0, but not both, a parabola.
| Quote: | I have also seen the general form most commonly expressed as
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Would it serve me better to concentrate on this equation?
|
Probably not.
Though it is possible to tell the type of conic from the values of A, B,
and C for such equations.
For B^2 -4*A*C < 0, an ellipse (circle for A=C and B=0 only),
for B^2 - 4*A*C = 0, a parabola,
for B^2 - 4*A*C > 0, an hyperbola. |
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uberclay
science forum beginner
Joined: 22 May 2006
Posts: 7
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| Posted: Sun Jun 18, 2006 8:03 am Post subject:
Re: Understanding the general from of a conic as it relates to standard form equations - Grade 11
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| Quote: | My text states the general form of conics as
Ax^2 + By^2 + Fx + Gy + C = 0
Actually this is only true for conics that have an axis of symmetry
parallel to a coordinate axis.
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I see - those are all we've done so far.. |
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matt271829-news@yahoo.co.
science forum Guru
Joined: 11 Sep 2005
Posts: 846
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| Posted: Sun Jun 18, 2006 10:16 am Post subject:
Re: Understanding the general from of a conic as it relates to standard form equations - Grade 11
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Virgil wrote:
| Quote: | In article <1150609937.105338.286630@c74g2000cwc.googlegroups.com>,
"uberclay" <david.palmatier@gmail.com> wrote:
I am having a hard time understanding the general form of conics as it
maps to the individual equations in standard form (2D)
Using the following standard forms
(x - h)^2 = 4p(y - k) is a parabola [P]
(x - h)^2 + (y - k)^2 = r^2 is a circle [C]
[(x - h)^2/a^2] + [(y - k)^2/b^2] = 1 or [(x - h)^2/b^2] + [(y -
k)^2/a^2] = 1 is an ellipse [E]
[(x - h)^2/a^2] - [(y - k)^2/b^2] = 1 or [(y - k)^2/a^2] - [(x -
h)^2/b^2] = 1 is a hyperbola [H]
I derive
[P] x^2 - 2hx + h^2 - 4py + 4pk = 0
[C] x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0
[E] a^2 x^2 - a^2 2hx + a^2 h^2 + b^2 y^2 - b^2 2ky + b^2 k^2 - a^2 b^2
= 0
[E] b^2 x^2 - b^2 2hx + b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2 b^2
= 0
[H] a^2 x^2 - a^2 2hx + a^2 h^2 - b^2 y^2 + b^2 2ky - b^2 k^2 - a^2 b^2
= 0
[H] - b^2 x^2 + b^2 2hx - b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2
b^2 = 0
My text states the general form of conics as
Ax^2 + By^2 + Fx + Gy + C = 0
Actually this is only true for conics that have an axis of symmetry
parallel to a coordinate axis.
The more general conic equation is of form
A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0
for suitable constants A, B, C, D, E, AND F.
For equations of form A*x^2 + B*y^2 + F*x + G*y + C = 0,
if A = B <> 0, you have a circle,
if A*B > 0, an ellipse,
if A*B < 0, an hyperbola, and
if A = 0 or B = 0, but not both, a parabola.
I have also seen the general form most commonly expressed as
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Would it serve me better to concentrate on this equation?
Probably not.
Though it is possible to tell the type of conic from the values of A, B,
and C for such equations.
For B^2 -4*A*C < 0, an ellipse (circle for A=C and B=0 only),
for B^2 - 4*A*C = 0, a parabola,
for B^2 - 4*A*C > 0, an hyperbola.
|
To be pedantic, this is not strictly true as there are various
degenerate cases also to consider: imaginary ellipse, crossed lines,
parallel lines, single line; imaginary lines, possibly others I
forgot... |
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Virgil
science forum Guru
Joined: 24 Mar 2005
Posts: 5536
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| Posted: Sun Jun 18, 2006 5:13 pm Post subject:
Re: Understanding the general from of a conic as it relates to standard form equations - Grade 11
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In article <1150625765.899502.124250@c74g2000cwc.googlegroups.com>,
matt271829-news@yahoo.co.uk wrote:
| Quote: | Virgil wrote:
In article <1150609937.105338.286630@c74g2000cwc.googlegroups.com>,
"uberclay" <david.palmatier@gmail.com> wrote:
I am having a hard time understanding the general form of conics as it
maps to the individual equations in standard form (2D)
Using the following standard forms
(x - h)^2 = 4p(y - k) is a parabola [P]
(x - h)^2 + (y - k)^2 = r^2 is a circle [C]
[(x - h)^2/a^2] + [(y - k)^2/b^2] = 1 or [(x - h)^2/b^2] + [(y -
k)^2/a^2] = 1 is an ellipse [E]
[(x - h)^2/a^2] - [(y - k)^2/b^2] = 1 or [(y - k)^2/a^2] - [(x -
h)^2/b^2] = 1 is a hyperbola [H]
I derive
[P] x^2 - 2hx + h^2 - 4py + 4pk = 0
[C] x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0
[E] a^2 x^2 - a^2 2hx + a^2 h^2 + b^2 y^2 - b^2 2ky + b^2 k^2 - a^2 b^2
= 0
[E] b^2 x^2 - b^2 2hx + b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2 b^2
= 0
[H] a^2 x^2 - a^2 2hx + a^2 h^2 - b^2 y^2 + b^2 2ky - b^2 k^2 - a^2 b^2
= 0
[H] - b^2 x^2 + b^2 2hx - b^2 h^2 + a^2 y^2 - a^2 2ky + a^2 k^2 - a^2
b^2 = 0
My text states the general form of conics as
Ax^2 + By^2 + Fx + Gy + C = 0
Actually this is only true for conics that have an axis of symmetry
parallel to a coordinate axis.
The more general conic equation is of form
A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0
for suitable constants A, B, C, D, E, AND F.
For equations of form A*x^2 + B*y^2 + F*x + G*y + C = 0,
if A = B <> 0, you have a circle,
if A*B > 0, an ellipse,
if A*B < 0, an hyperbola, and
if A = 0 or B = 0, but not both, a parabola.
I have also seen the general form most commonly expressed as
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Would it serve me better to concentrate on this equation?
Probably not.
Though it is possible to tell the type of conic from the values of A, B,
and C for such equations.
For B^2 -4*A*C < 0, an ellipse (circle for A=C and B=0 only),
for B^2 - 4*A*C = 0, a parabola,
for B^2 - 4*A*C > 0, an hyperbola.
To be pedantic, this is not strictly true as there are various
degenerate cases also to consider: imaginary ellipse, crossed lines,
parallel lines, single line; imaginary lines, possibly others I
forgot...
|
Given that one has a real conic, the analysis is correct. Since the OP
was not considering degenerate cases for Ax^2 + By^2 + Fx + Gy + C = 0,
I saw no reason to bring them up for the more general situation. |
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