Graph and Differential equations for hyperbolas

shayaan_musta

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.

1mmorta1

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

shayaan_musta

Oh thanks. It is quite helpful.

shayaan_musta

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

Is it?

HallsofIvy

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.

shayaan_musta

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?

HallsofIvy

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.

shayaan_musta

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