Normal form of the conic section equation

In summary, the conversation discusses the normal form of the conic section equation, which describes a circle and can also be used to represent other conic sections such as hyperbolas. The term "normal" may be used to differentiate between different forms of equations. The conversation also delves into the conversion of equations, such as converting xy=c to x²-y²+c=0, and the use of rotation equations to transform the axes. However, these equations are not used for actual rotation but rather for conversion between different forms of equations.
  • #1
PainterGuy
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TL;DR Summary
Why this equation, x²+y²+2ax+2by+c=0, is called 'normal form of the conic section equation'.
Hi,

The following is called normal form of the conic section equation:
x²+y²+2ax+2by+c=0

A circle is one of the conic sections when considered as a special of ellipse. I'm confused as to why the the given equation is called "normal form of the conic section equation" when, in my opinion, the equation only describes a circle, a point which could be said to be a special case of circle, imaginary locus or imaginary circle; as I understand it the equation has nothing to do with the conic sections other than the circle which is a special case. Could it represent any other conic section?

The shown below is equation in its context:

?hash=3ec9a72c83dc9a5c9e2ce1c60870a5f9.jpg


I have tried below to relate the "normal form of the conic section equation" to general equation of a circle with radius R and center at (x₀, y₀).

?hash=3ec9a72c83dc9a5c9e2ce1c60870a5f9.jpg


Thank you for your help and time!
 

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  • #2
I don't know why the term "normal" is used. Your analysis agrees with what is given in the extract. Essentially, it's a circle.
 
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  • #3
A and C both may = 0 as long as B is not. (hyperbola)
 
  • #4
Thank you!

mathman said:
A and C both may = 0 as long as B is not. (hyperbola)

I'm sorry but I don't follow your point. Could you please elaborate a bit? Thanks.

?hash=d73e364b9a54376f63a4ab55d4b63554.jpg
 

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  • #5
A little more generally, equation forms might be called "general", "standard", or other kinds ("vertex form"). Why then need to use any designation of "normal"? Maybe to know if an equation is general or standard or something else and why, might be good.
 
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  • #6
##xy+F=0## is an equation for a hyperbola.
 
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  • #7
Thank you!

mathman said:
##xy+F=0## is an equation for a hyperbola.

Thanks for pointing this out that xy+F=0 represents a hyperbola but how can one derive "xy+F=0" from this so-called 'normal form of the conic section equation' x²+y²+2ax+2by+c=0? I don't see a way.

By the way, another related question is that how one can convert the general equation of hyperbola, (x-x₀)²/a² - (y-y₀)²/b² = 1, into xy=c or xy+c=0.

symbolipoint said:
A little more generally, equation forms might be called "general", "standard", or other kinds ("vertex form"). Why then need to use any designation of "normal"? Maybe to know if an equation is general or standard or something else and why, might be good.

I'd say that many a time terminology and definitions vary from one source to another and one should not worry oneself with the terminology unless it seems really important. I tried to find the answer about the difference between 'standard form', 'normal form', 'general form', etc. and got confused so left it halfway because it didn't seem much important to me.

Anyway, some of the links given below might be useful.

1: https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/simple.html
2: https://elsenaju.eu/Calculator/normal-form-to-vertex-form.htm
3: watch?v=cjXafzJ-gEk (add "https://www.youtube.com/" before "watch")
4: https://arxiv.org/pdf/1702.05789.pdf
5: https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html
6: https://www.physicsforums.com/threads/general-form-vs-standard-form-of-a-line.640797/
7: https://www.mathsisfun.com/algebra/quadratic-equation.html
8: https://www.analyzemath.com/quadraticg/quadraticg.htm
9: watch?v=-r4v9J0tn-o (add "https://www.youtube.com/" before "watch")
 
  • #8
Look at what actual textbooks say for the kinds of or names of the equations' forms.
 
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  • #9
To convert ##xy=c##, set ##x=x'+y'## and ##y=x'-y'## to get ##x'^2-y'^2+c=0##.
 
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  • #10
Thank you!

mathman said:
To convert ##xy=c##, set ##x=x'+y'## and ##y=x'-y'## to get ##x'^2-y'^2+c=0##.

xy=c --- equation 1
using x=x′+y′ and y=x′-y′
(x′+y′)(x′-y′)=c
(x′)²-x′y′+y′x′-(y′)²=c
(x′)²-(y′)²=c
letting x′=X, y′=Y
X²-Y²=c --- equation 2

Now let's graph the equations.

Graph #1:
xy=c
⇒y=c/x
let c=2
y=2/x

?hash=fcd193a231171e92ff8264d6a8bab7d1.jpg


Graph #2:
x²-y²+c=0
⇒-y²=-x²-c
⇒y²=x²+c
⇒y=√(x²+c)
let c=2
⇒y=√(x²+2)

?hash=fcd193a231171e92ff8264d6a8bab7d1.jpg


It's clear that both equations represent a hyperbola and one equation is rotated version of another. It's 45 degrees rotation.

These two equations, x=x′+y′ and y=x′-y′, are used to rotate the axes by 45 degrees. But I don't understand where these equations come from. The rotation of axes is given as follows.

?hash=fcd193a231171e92ff8264d6a8bab7d1.jpg


Could you please help me with this? Thank you!
 

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  • #11
Hi,

Could someone please help me with the query from my last post?

The query was "These two equations, x=x′+y′ and y=x′-y′, are used to rotate the axes by 45 degrees. But I don't understand where these equations come from."

Thank you!
 
  • #12
Hi,

In post #10 above I made few mistakes so in this posting I would try to fix them.

The equation of a hyperbola in rectangular coordination is given as:

?hash=b16c846c00d2c40c4db3de006e092243.jpg


where (x₀,y₀) is center and "a" is distance from center to vertex and "b" is vertical distance from vertex to the asymptote, or distance from center to co-vertex. The given equation encapsulates definition of a hyperbola and assumes that the major axis is parallel to x-axis. For example, there cannot be "-1", "2", or "2.2" on the right side of equation. It can only be "1" because that's what the definition, i.e. equation, says.

245425


Also, if the major axis is parallel to y-axis instead, one can interchange x and y in the equation above. Sometimes one can find a conic section, including hyperbola, rotated and in such cases one can use transformation equations involving pure rotation shown below to find the final equation.

x′=xcos(α)+ysin(α)
y′=ycos(α)-xsin(α)

We know from post #10 that hyperbola represented by xy=c is a 45 degrees rotated version.

?hash=b16c846c00d2c40c4db3de006e092243.jpg


These two equations, x=x′+y′ and y=x′-y′, do not actually represent rotation; they are more of a conversion tool from one form to another.

Thank you!
 

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Last edited:

Related to Normal form of the conic section equation

What is the normal form of the conic section equation?

The normal form of the conic section equation is a simplified form of the general equation of a conic section, where the coefficients of the x^2 and y^2 terms are equal and the constant term is zero.

What are the different types of conic sections?

The four types of conic sections are parabolas, circles, ellipses, and hyperbolas. Each type has a unique shape and can be described by a specific equation.

How do you identify the type of conic section from its equation in normal form?

To identify the type of conic section from its equation in normal form, you can look at the coefficients of the x^2 and y^2 terms. If they are equal, the conic section is a circle or an ellipse. If they have opposite signs, the conic section is a hyperbola. If one of the coefficients is zero, the conic section is a parabola.

What does the value of the coefficient of the x^2 or y^2 term represent in the normal form of the conic section equation?

The value of the coefficient of the x^2 or y^2 term represents the shape of the conic section. A positive coefficient indicates a circle or an ellipse, while a negative coefficient indicates a hyperbola. A zero coefficient indicates a parabola.

How is the normal form of the conic section equation used in real-world applications?

The normal form of the conic section equation is used in various fields of science, engineering, and mathematics to model and analyze real-world phenomena. For example, it can be used to describe the orbits of planets and satellites, the shape of lenses and mirrors in optics, and the paths of projectiles in physics.

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