how to find the tangent space of 1D manifold in $\mathbb{R}^3$?

Question

I want to find the tangent space of $M$ in $p$, where M is defined by $z=x^2+y^2$ and $z=4-y$ and $p=(2,-1,5)$.

I know how to find tangent spaces of manifolds which are defined using one equation - I find function f and value a such that M = $f^{-1}(a)$ and than the tangent space equals $\text{ker}(df_p)$. here I don't find such $f$, and don't know how to continue.

Answer 1

Hint

Your manifold is defined as the intersection of two sub manifolds.

The tangent space (a line here) is the intersection of the tangent spaces (planes here) at $p$.