Parameterize the intersection of the surfaces

[andyk23]

Parameterize the intersection of the surfaces z=x^2-y^2 and z=x^2+xy-1

What's getting me stuck on this problem is the xy. I set x=t

z=x^2-y^2
z=t^2-y^2

z=x^2+xy-1
t^2-y^2=t^2+ty-1
y^2=1-ty
Thats as far as of come, I'm stuck on this

 

[Dick]

Parameterize the intersection of the surfaces z=x^2-y^2 and z=x^2+xy-1

What's getting me stuck on this problem is the xy. I set x=t

z=x^2-y^2
z=t^2-y^2

z=x^2+xy-1
t^2-y^2=t^2+ty-1
y^2=1-ty
Thats as far as of come, I'm stuck on this

That looks ok so far. So now you want to solve y^2=1-ty for y in terms of t, right? It's a quadratic equation. You should be able to solve that.

 

[andyk23]

That's the part I'm stuck on, y^2+ty-1=0; (y+?)(y-?)=0

 

[Dick]

That's the part I'm stuck on, y^2+ty-1=0; (y+?)(y-?)=0

You don't really directly solve something like this by factoring. Do you know the quadratic formula?

 

[andyk23]

sorry I was thinking something else for the quadratic eqn y= -t+/- sq rt(t^2=4)/2

 

[Dick]

sorry I was thinking something else for the quadratic eqn y= -t+/- sq rt(t^2=4)/2

That would be right, if you could clean up the formatting.

 

[andyk23]

then r(t)=<t, (-t+/- sq rt(t^2=4))/2, t^2-(-t+/- sq rt(t^2=4))/2> thanks for your help!

 

[Dick]

then r(t)=<t, (-t+/- sq rt(t^2=4))/2, t^2-(-t+/- sq rt(t^2=4))/2> thanks for your help!

Gack. sq rt(t^2=4)? What's that supposed to mean? And your z component is wrong. But I think you can clean this up on your own.

 

[andyk23]

would the Z component be z= x^2-y^2 and just put the x and y in?

 

[Dick]

would the Z component be z= x^2-y^2 and just put the x and y in?

It would probably be easier to put them into z=x^2+xy-1 so you don't have to square y, don't you agree?

 

[andyk23]

Agreed, thanks again for all your help!