tangent hyperplanes and tanget planes, a bit confused about dimensions

I thought I understood tangent planes until I saw tangent hyperplane and 3-d contour here. So a little bit of help will be most welcome. (i) and (ii) is no problem and agreed to the textbook answer. I am a bit stuck on (iii).

Question (Calculus, Binmore and Davies, 3.9(#5*))

Let

i. Find the gradient of f at the point

ii. Find the tangent hyperplane to the hypersurface

where .

iii. Find the normal and the tangent plane to the contour

at .

Answer.

i.

ii. To find tangent hyperplane, I want to use the formula

iii. To me, the equation for the countour looks exactly the same as the equation to the surface in (ii) as u=0 in both cases.

*I understand in (ii) the hypersurface is in 4 dimentional space, and the hyperplane is in 3 dimensions. Here the contour is in 3 dimensions (??) and the tangent plane is 2-dimentional. *

*???*

I am not sure what to do, but *I was thinking may be try to express z as the function of x and y and apply the same template as in (ii) ie*

* *

*???*