first of all, what is the difference between the tangent space and the tangent plane? I tried to find the tangent space of the hyberpoloid $$x^2 +y^2 -z^2 =a$$ , $$a>0 $$ at the point $$(\sqrt{a},0,0)$$ in this way: $$f(x,y,z)=x^2 +y^2 -z^2$$ $$f_{x}=2x$$, $$f_{y}=2y$$, $$f_{z}=-2z$$
then, at the point $$(2\sqrt{a},0,0)$$ we get $$(2\sqrt{a},0,0)$$ hence the tangent space is $$2\sqrt{a}(x-\sqrt{a}) +0(x-0) -0(x-0)=0$$
$$x=\sqrt{a}$$
I feel that something is wrong.Can you help me please?