Can anyone help me with hyperbolas

I

ILoveMath4Days

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I was playing with the hyperbola (x)^2 /9 - (y)^2 /9, and when I multiplied them together instead of subtracting them it gave me something weird that I can't quite explain.
I could not send an image of the graph but I can put the demos link, https://www.desmos.com/calculator/cv3milgu6c, please do not mess with the values but you can turn lines on and off. If someone could take a look at it and explain why when you multiply the parts of the hyperbola the give you the thing that I can't explain, and by dividing it gives you the asymptotes of the hyperbola, that would be greatly appreciated.
 
Hyperbolas centered at the Origin look like this: ax^2 + bxy + cy^2 = d (with appropriate relationships between a, b, and c.)

There is no sense in expecting another hyperbola if you divide two of the pieces. ax^2 / cy^2 is what? Where did the rest of it go? It is not particularly related to the hyperbola where you started. It's just another thing and it may not be helpful to think that you started with an hyperbola.

There is no sense in expecting another hyperbola if you multiply two of the pieces. ax^2 * cy^2 is what? Where did the rest of it go? It is not particularly related to the hyperbola where you started. It's just another thing and it may not be helpful to think that you started with an hyperbola. I might expect it to be somewhat reminiscent of an hyperbola, but since it is now a 4th degree term, it should not behave like a sum of quadratic terms.

It might be helpful to know exactly what it is you are trying to do.
 
I want to figure out why the terms did what they did when I multiplied and divided them. Also did you take a look at the desmos link, it should tell you what I am talking about.
 
First, you are aware that the hyperbola you are starting with is not [MATH]\frac{x^2}{9} - \frac{y^2}{9}[/MATH], but [MATH]\frac{x^2}{9} - \frac{y^2}{9} = 1[/MATH]. The former is an expression, not an equation, and it has a value, not a graph.

What you get when you divide the terms is equivalent to the equation [MATH]\frac{x^2}{9} - \frac{y^2}{9} = 0[/MATH], replacing 1 with 0. If you think about it, you can see why this gives the asymptotes; in effect, this is the limit when you divide x and y in the original equation by n and let n approach infinity.

What you got when you multiplied is not really related to the original, but it is equivalent to [MATH]|y| = \frac{1}{|x|}[/MATH] (taking the square root of both sides). This is your rectangular hyperbola [MATH]y = \frac{1}{x}[/MATH] stretched by a factor of 3 in both directions, together with its reflection.
 
One thing that I am still confused about is that when the 4 other lines were created by multiplying each term and having them equal 1, the line of symmetry of each new line is the asymtoe of the original hyperbola.
 
ILoveMath4Days said:
sorry that is what I meant, x^2/9 - y^2/9 =1
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Yes, I said you were aware, since that's what you put in Desmos. ;)
ILoveMath4Days said:
So what you are saying is that the four unrelated lines are the inverse of the hyperbola.
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What four lines? I only see two straight lines. Do you mean the four branches of the green curve? Those aren't unrelated.
What is the inverse of a curve? Do you mean the reciprocal function y = 9/x? Are you aware that that is a (rotated) hyperbola?
ILoveMath4Days said:
One thing that I am still confused about is that when the 4 other lines were created by multiplying each term and having them equal 1, the line of symmetry of each new line is the asymptote of the original hyperbola.
Click to expand...
Why is that confusing, rather than just interesting?
 
Thank you for all the help and the 4 green lines are the rotated form of the hyperbola, I went through each equation and I now understand why the asymtotes are created when you divide and the rotated hyperbolas are created when you multiply.
 
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