= Solution
Let
$$
F(x)=-x_0^2+x_1^2+x_2^2+x_3^2=x^TMx,
\qquad
M=\operatorname{diag}(-1,1,1,1).
$$
At $x\in F^{-1}(-1)$,
$$
dF_x(v)=2x^TMv.
$$
This differential is nonzero because $x\ne0$, so $-1$ is a <regular value>. The <regular level set theorem> makes $F^{-1}(-1)$ a three-dimensional embedded submanifold; intersecting it with the open set $\{x_0>0\}$ preserves that property. Therefore $X$ is the <three-dimensional hyperboloid model>, and
$$
\boxed{
T_xX=\ker dF_x
=\{v\in\mathbb R^4:-x_0v_0+x_1v_1+x_2v_2+x_3v_3=0\}
=\{v:x^TMv=0\}.
}
$$
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