= Solution
With $\omega=B/m$, magnetic force does no work, so $|\dot x|$ is constant; its zero $z$ component also makes $\dot z$ constant. For the stated data,
$$
\dot x=u\cos\omega t,\qquad \dot y=-u\sin\omega t,
$$
$$
x=\frac u\omega\sin\omega t,\qquad y=\frac u\omega(\cos\omega t-1).
$$
This is a circle of radius $u/\omega$ centred at $(0,-u/\omega)$.
With $\alpha=\mu/m$, the complex <velocity> is $\dot x+i\dot y=u e^{-(\alpha+i\omega)t}$. Hence
$$
x=\frac{u[\alpha(1-e^{-\alpha t}\cos\omega t)+\omega e^{-\alpha t}\sin\omega t]}{\alpha^2+\omega^2},
$$
$$
y=\frac{u[\alpha e^{-\alpha t}\sin\omega t-\omega(1-e^{-\alpha t}\cos\omega t)]}{\alpha^2+\omega^2}.
$$
The path spirals into $u(\alpha,-\omega)/(\alpha^2+\omega^2)$. As $u$ varies, this lies on $y=-(\omega/\alpha)x$, at distance
$$
\frac u{\sqrt{\alpha^2+\omega^2}}=\frac{mu}{\sqrt{\mu^2+B^2}}
$$
from the origin.
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