= Solution
Translate the horizontal coordinate by $x$. Then $X_t+x$ is a standard <Brownian motion> starting at zero, and continuity identifies $\tau$ with its <first-passage time> to $x$. The vertical coordinate remains an independent standard <Brownian motion> starting at zero. Part (a) therefore gives the <Cauchy exit law from a Brownian half-plane>:
$$
\boxed{Y_\tau\overset d=xC,\qquad f_{Y_\tau}(y)=\frac{x}{\pi(x^2+y^2)}.}
$$
The TeX transcription's initial value $-z$ and later $Y_T$ are both corrected here to the PDF's $-x$ and $Y_\tau$.
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