Let $S_r=\{se^{i\theta}:0<s\leq r\}$. Since $S_r=rS_1$, the <scaling and translation of half-plane capacity> gives
$$
\operatorname{hcap}(S_r)=r^2\operatorname{hcap}(S_1)=ar^2.
$$
The <half-plane-capacity parameterization> requires $ar(t)^2=2t$, so
$$
r(t)=\sqrt{\frac{2t}{a}},
\qquad
\gamma(t)=\sqrt{\frac{2t}{a}}e^{i\theta}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 1 / c / Solution

{title2=$\operatorname{hcap}(K_t)=2t$}

A growing hull is parameterized by half-plane capacity when $\operatorname{hcap}(K_t)=2t$. The factor two makes its <Chordal Loewner equation> take the conventional form $\partial_tg_t=2/(g_t-U_t)$.

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