Graph and Differential equations for hyperbolas

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Discussion Overview

The discussion centers around the equation xy=c, which represents a hyperbola. Participants explore how to derive multiple graphs from this single equation and the corresponding differential equations. The conversation also touches on the relationship between hyperbolas and parabolas.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that y = c/x represents a family of hyperbolas that varies with different values of c.
  • Another participant questions the relationship between the hyperbola and the equation y^2 = 4ax, suggesting both are conic sections.
  • Some participants clarify that y = c/x is equivalent to the original hyperbola equation xy = c.
  • There is a suggestion that the family of hyperbolas can be plotted by varying c, including both positive and negative values.
  • A participant raises a question about whether y = 0 can be considered a parabola, leading to further inquiry about the nature of parabolas and their definitions.

Areas of Agreement / Disagreement

Participants generally agree on the equivalence of y = c/x and xy = c as representations of hyperbolas. However, there is some uncertainty regarding the classification of y = 0 and its relation to parabolas, indicating a lack of consensus on that point.

Contextual Notes

Participants express varying assumptions about the nature of conic sections and the implications of different values of c, but these assumptions remain unresolved.

shayaan_musta
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Hello experts!
Hope all of you will be fine.

I have an equation i.e. xy=c
And we all know it is hyperbola.

Now I say "graph some of the hyperbolas xy=c". Then kindly tell me how can we extract more than 1 graph from this single equation? And you will write the differential equations for them. while here only 1 hyperbola is given i.e. xy=c.


If you have any confusion about the question the kindly tell me. I will try to clear more.

Thanks in advance.
 
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Assuming c is some constant, you have y = c/x. This is a family of graphs, which varies based on values of c. I.E. y = 1/x, y = 2/x, ...y = c/x
 
1mmorta1 said:
Assuming c is some constant, you have y = c/x. This is a family of graphs, which varies based on values of c. I.E. y = 1/x, y = 2/x, ...y = c/x

Oh thanks. It is quite helpful.
 
y[itex]^{2}[/itex]=4ax is also a parabola & and y=[itex]\frac{c}{x}[/itex] too?

Is it?
 
Yes, [itex]y^2= 4ax[/itex] would be a family of parabolas, all passing through (0, 0) having different foci.

I'm not sure what your question about y= c/x is. It is the same as xy= c, your original hyperbola system.
 
HallsofIvy said:
Yes, [itex]y^2= 4ax[/itex] would be a family of parabolas, all passing through (0, 0) having different foci.

I'm not sure what your question about y= c/x is. It is the same as xy= c, your original hyperbola system.

My real question as you can see that, how can you plot some hyperbolas families from general equation i.e. xy=c?

This could be y=c/x and therefore some families will be y=1/x, y=2/x, y=3/x...so on.
Where c=any arbitrary constant.

Am I right?
 
Yes, that is exactly what it is saying. They will be parabolas having the x and y axes as asymptotes, passing through (1, c) and (-1, -c), for each number c. Be sure to include some values of c negative and c= 0.
 
HallsofIvy said:
Yes, that is exactly what it is saying. They will be parabolas having the x and y axes as asymptotes, passing through (1, c) and (-1, -c), for each number c. Be sure to include some values of c negative and c= 0.

As you said c=0 this implies that y=0/x or y=0
Can y=0 be a parabola? Is it so?
 

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