= Solution
The answer is no. Let $a_i\ne0$ tend to zero and take
$$
p_i=(a_i,0,0,0)\in X_{a_i},
\qquad p_i\to p=(0,0,0,0)\in X_0.
$$
At a point $x\in X_a$ with $a\ne0$, differentiating the defining equations gives
$$
x_1w_1+2x_2w_2=0,
\qquad w_3=aw_4.
$$
Thus
$$
w_i=(0,1,0,0)\in T_{p_i}X_{a_i}
$$
for every $i$, and $w_i\to w=(0,1,0,0)$. But $X_0$ is the $x_4$-axis, so
$$
T_pX_0=\operatorname{span}\{(0,0,0,1)\},
\qquad w\notin T_pX_0.
$$
Hence
$$
\boxed{\ T_pX_a\text{ does not depend continuously on }(p,a)\text{ at }a=0.\ }
$$
The obstruction is the <dimension jump in a smooth family of manifolds>.
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