Solving Tangent Plane and Perpendicular Plane at (1,1,1)

[brads]

Got a problem which should be easy but having trouble...

"Find the equation of the tangent plane to z=f(x,y)=x^2 + y^2 - 1 at point (1,1,1)
"Find an equation to a plane that is perpendicular to that tangent plane and also passes through the point (1,1,1)

Thanks!

 

[arildno]

Welcome to PF!

Are you familiar with the theorem that the gradient of a surface G(x,y,z)=0 is normal to that surface?
Use that hint (G(x,y,z)=z-f(x,y)).

 

[brads]

I figured out the first part of the problem the tangent plane to f(x,y) is 2x+2y-z-3=0. The normal vector of that plane is thus <2,2,-1>. I am lost on how to solve the next part, I keep trying different equations but coming up with too many variables.

 

[arildno]

There's an infinity of planes which are normal to the tangent plane.
Pick one of them; the requirement is merely that some tangent vector in the tangent plane should be the normal to the plane you choose.
(the normal of the tangent plane is to be a tangent vector in the plane you choose)

 

[Gokul43201]

"too many variables" only allows you to set one of them to any convenient value. Like arildno said, there are an infinite number of normal planes passing through a given point; so you get to pick anyone you like.