How to find the straight tangent line?

[Helloooo]

Homework Statement

Find the straight tangent line at (2,-1) to the level curve of f(x,y)=x^2-y^2

Relevant Equations

f(x,y)=x^2-y^2
Point: (2,-1)

I have solved the gradient:

gradf(2,-1)=(4,2)

and have the tangent plane:

4x+2y+3=0

Somehow the answer is:

3=2x+y

And i really don´t understand why.

 

[Orodruin]

and have the tangent plane:

4x+2y+3=0

No you don’t. Inserting (2,-1) in the LHS gives 4*2+2(-1) = 8-2 = 6 so the required point is not on that plane.

 

[Helloooo]

No you don’t. Inserting (2,-1) in the LHS gives 4*2+2(-1) = 8-2 = 6 so the required point is not on that plane.

Firstly i noticed an error in my writing.
The tangent plane i got was 4x+2y-3=0.
I´m sorry for that
So i redid the tangent plane to
4x+2y-6=0
However i still don´t know how to fint the straight tangent line.
Thank you

 

[Orodruin]

However i still don´t know how to fint the straight tangent line.

That's it, right there:

4x+2y-6=0

 

[Helloooo]

That's it, right there:

Does that mean that the tangent plane and straight tangent line is the same?
If so, why are they called differently or is it just in this particulary problem?

 

[Orodruin]

Does that mean that the tangent plane and straight tangent line is the same?
If so, why are they called differently or is it just in this particulary problem?

In two dimensions, a tangent line and the tangent plane are the same thing. In n dimensions the tangent plane has n-1 dimensions. It is a more general concept because it is not restricted to two dimensions.

 

[Helloooo]

In two dimensions, a tangent line and the tangent plane are the same thing. In n dimensions the tangent plane has n-1 dimensions. It is a more general concept because it is not restricted to two dimensions.

I see, thank you so much for your help!