Graph and Differential equations for hyperbolas

In summary, we can extract multiple graphs from the single equation xy=c, where c is a constant. These graphs will be hyperbolas with varying values of c. Similarly, for y^2=4ax, we can extract a family of parabolas with different foci. The equation y=c/x also represents a family of hyperbolas with the x and y axes as asymptotes, passing through (1,c) and (-1,-c) for each value of c. This also applies to values of c being negative and zero, with y=0 being a horizontal asymptote in the case of c=0.
  • #1
shayaan_musta
209
2
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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  • #2
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
 
  • #3
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.
 
  • #4
y[itex]^{2}[/itex]=4ax is also a parabola & and y=[itex]\frac{c}{x}[/itex] too?

Is it?
 
  • #5
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.
 
  • #6
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?
 
  • #7
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.
 
  • #8
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?
 

1. What is a hyperbola?

A hyperbola is a type of conic section that is formed by the intersection of a plane with a double cone. It is a symmetrical open curve with two branches that are mirror images of each other.

2. How is a hyperbola represented graphically?

A hyperbola can be graphed on a coordinate plane by plotting the center point, determining the vertices, and drawing the asymptotes. The equation of a hyperbola can also be used to plot points and draw the curve.

3. What is the standard form of a hyperbola?

The standard form of a hyperbola is (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) is the center of the hyperbola, a is the distance from the center to the vertices on the x-axis, and b is the distance from the center to the vertices on the y-axis.

4. How are differential equations used to study hyperbolas?

Differential equations can be used to study the properties of hyperbolas, such as their asymptotes, vertices, and foci. They can also be used to model real-life situations involving hyperbolas, such as the trajectory of a satellite orbiting a planet.

5. What is the relationship between a hyperbola and its asymptotes?

The asymptotes of a hyperbola are the lines that the curve approaches but never touches. The equation of the asymptotes can be found by using the slope formula and the center of the hyperbola. The asymptotes also determine the shape and orientation of the hyperbola.

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