An affine hyperplane $H$ in $\mathbb{R}^{3}$ is an affine subspace in $\mathbb{R}^{3}$ of dimension 2, i.e. there exists a $h \in \mathbb{R}^{3}$ and a two-dimensional $\mathbb{R}$ vector subspace $U$ in $\mathbb{R}^{3}$ with $H=h+U$. Consider the circular cone in $\mathbb{R}^{3}$:
K=$\{(x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} \mid x_{0}^{2}- x_{1}^{2}-x_{2}^{2}=0\}$
(a) Determine an affine hyperplane $H$ in $\mathbb{R}^{3}$, for which $H \cap K$ is a circle in $H$.
(b) Determine an affine hyperplane $H$ in $\mathbb{R}^{3}$ for which $H \cap K$ is a hyperbola.
[So explicitly, in the above notation for hyperplanes: They should be $U$ and $h$ and a $\mathbb{R}$ vector space isomorphism $j: U \cong \mathbb{R}^{2}$ so that for $H=h+U$ the following applies: $H \cap K=h+j^{-1}(D)$ with $D=\{\left(y_{1}, y_{2}\right) \in \mathbb{R}^{2} \mid y_{1}^{2}+y_{2}^{2}=1\}$.]
My attempt to (a) is:
We consider the cone: $x_{0}^{2}-x_{1}^{2}-x_{2}^{2}=0$ and set $x_{0}=1$ $\Rightarrow 1 =x_{1}^{2}+x_{2}^{2}$
and thus $U:=\{\left(\begin{array}{l}1 \\ x_{2} \\ x_{2}\end{array}\right) \in \mathbb{R}^{3}: x_{1}^{2} +x_{2}^{2}=1\}$ and $h=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$. Let $j: U \cong \mathbb{R}^{2}$ be the identity map, then we have
$H \cap K=h+j^{-1}(D)$
where $D=\{\left(y_{1}, y_{2}\right) \in \mathbb{R}^{2} \mid y_{1}^{2}+y_{2}^{2}=1\}$.
I'm not sure about that, is it the correct way to approach this problem?
Thanks!