Parametric equations question: What are the parametric equations for the plane curve xy^4 = 7 ?

[Marie_fun_math]

Hello everyone!! I'm nearing the end of my class, and I felt like I was really getting the hang of this, but I am hoping to get a little help on this specific problem.



Any help is appreciated, thank you!!

 

[topsquark]

Hello everyone!! I'm nearing the end of my class, and I felt like I was really getting the hang of this, but I am hoping to get a little help on this specific problem.

View attachment 36309

Any help is appreciated, thank you!!

This is a badly worded question, and does not have a unique answer.

We may choose our parameterization. For example, if we simply set
x = s then [imath]y = \left ( \dfrac{7}{s} \right )^{1/4}[/imath]

If we set
[imath]x = \dfrac{7}{s^2}[/imath], then [imath]y = \pm \sqrt{s}[/imath]

The idea is to decide on what to call either of the variables in terms of s, then solve for the other one.

-Dan

 

[skeeter]

I would start by letting [math]y(s) = s[/math]

 

[skeeter]

note ...

[imath]xy^4 = 7 \implies x > 0 \text{ and } y \ne 0[/imath]

 

[Steven G]

This is a badly worded question, and does not have a unique answer.

We may choose our parameterization. For example, if we simply set
x = s then [imath]y = \left ( \dfrac{7}{s} \right )^{1/4}[/imath]

If we set
[imath]x = \dfrac{7}{s^2}[/imath], then [imath]y = \pm \sqrt{s}[/imath]

The idea is to decide on what to call either of the variables in terms of s, then solve for the other one.

-Dan

Dan, do these type problem ever have a unique solution?

 

[Steven G]

xy⁴=7.
Now solve for x or y.

x=7/y⁴. Let y=s and then x=7/s⁴
OR
y = (7/x)^1/4. Let x=s and y = (7/s)^1/4

It really is that easy. For the record, in what I have above you can choose y = 3s⁵ and then x= 7/[(3s⁵)]⁴. The point is you can let y be any function of s. Of course, you should take the easy way out and just let y=s!

 

[khansaheb]

any function

Any function? Sine, Cosine, log - any function?

 

[topsquark]

Dan, do these type problem ever have a unique solution?

Yes, but the problem statement reads

What are the parametric equations for...

which implies to me that they mean it's unique.

-Dan

 

[Marie_fun_math]

Thanks so much, everyone!!?

 

[Steven G]

Any function? Sine, Cosine, log - any function?

Let's see.
xy⁴=7 which implies x=7/y⁴.
Let y= sin(s). Then x=7/sin⁴s

Let y= log(s). Then x=7/log⁴s

Do you have some problems with this? I spent zero time on domain issues but think that can easily be cleaned up.

 

[khansaheb]

Let's see.
xy⁴=7 which implies x=7/y⁴.
Let y= sin(s). Then x=7/sin⁴s

Let y= log(s). Then x=7/log⁴s

Do you have some problems with this? I spent zero time on domain issues but think that can easily be cleaned up.

Yes I do.

domain issues but think that can easily be cleaned up.

That is exactly the issue in this problem. It involves "multiplicative inverses" which splits the domains of trigonometric functions.