= Solution
The scaling rule for the <Chordal Loewner equation> shows that $A_t=\sqrt t\,A_1$, because $r^{-1}U_{r^2t}=a\sqrt t$. To identify the shape, rather than infer it from scaling alone, construct the inverse map explicitly. Put
$$
d=\sqrt{a^2+16},\qquad r_\pm=\frac{a\pm d}{2},\qquad
\alpha=-\frac{r_-}{d}\in(0,1).
$$
Then $r_-<0<r_+$, $r_-r_+=-4$, and $\alpha r_++(1-\alpha)r_-=0$. Define
$$
f_t(w)=(w-r_+\sqrt t)^\alpha(w-r_-\sqrt t)^{1-\alpha},
$$
using logarithms whose arguments lie in $(0,\pi)$ on $\mathbb H$. Its <Laurent series> is $f_t(w)=w-2t/w+O(w^{-2})$, so it has the inverse <hydrodynamic normalization at infinity>.
On the real interval $(r_-\sqrt t,r_+\sqrt t)$, the first factor has argument $\pi\alpha$ and the second has argument zero. The image therefore lies on one straight ray. Its modulus increases from zero to a maximum at $w=a\sqrt t$, and then decreases to zero. The complementary real intervals map to the negative and positive real axes. The <argument principle>, or the usual conformal slit-map construction, shows that $f_t$ maps $\mathbb H$ conformally onto the half-plane minus that segment; the interval traverses its two sides.
The boundary walk has winding number one around every point of the half-plane off the segment: the two traversals of the slit cancel, leaving the real boundary and a large semicircle. The <argument principle> therefore gives exactly one preimage of each such point.
It remains to check that this is the correct <Loewner chain>. For $f=f_1$,
$$
\frac{f'(w)}{f(w)}=\frac{w-a}{(w-r_+)(w-r_-)}
=\frac{w-a}{w^2-aw-4}.
$$
Together with $f_t(w)=\sqrt t\,f(w/\sqrt t)$, this gives
$$
\partial_tf_t(w)=-\frac{2f_t'(w)}{w-a\sqrt t},
$$
the inverse form of the <Chordal Loewner equation>. Uniqueness therefore identifies its slit with $A_t$. Its endpoint is $\sqrt t\,f_1(a)$, where $f_1(a)\ne0$ lies in $\mathbb H$. This proves that the <square-root Loewner driving function generates a straight slit> and that
$$
\boxed{\bigcup_{t\geq0}A_t=\{s f_1(a):s>0\},}
$$
a straight ray from the boundary point $0$ to infinity.
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