High School Normal form of the conic section equation

[PainterGuy]

TL;DR

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:

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).

Thank you for your help and time!

 

[PeroK]

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.

 

[mathman]

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

 

[PainterGuy]

Thank you!

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.

 

[symbolipoint]

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.

 

[mathman]

$xy+F=0$ is an equation for a hyperbola.

 

[PainterGuy]

Thank you!

$xy+F=0$ 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.

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")

 

[symbolipoint]

Look at what actual textbooks say for the kinds of or names of the equations' forms.

 

[mathman]

To convert $xy=c$, set $x=x'+y'$ and $y=x'-y'$ to get $x'^2-y'^2+c=0$.

 

[PainterGuy]

Thank you!

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

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

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.

Could you please help me with this? Thank you!

 

[PainterGuy]

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!

 

[PainterGuy]

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:

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.

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.

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!