= Solution
Differentiate the affine relation with respect to $x$ and $y$:
$$
au_x+bv_x=0,
\qquad
au_y+bv_y=0.
$$
Using the <Cauchy-Riemann equations>, the second equation becomes
$$
bu_x-av_x=0.
$$
Thus
$$
\begin{pmatrix}a&b\\b&-a\end{pmatrix}
\begin{pmatrix}u_x\\v_x\end{pmatrix}=0.
$$
The <determinant> is $-(a^2+b^2)\ne0$, so $u_x=v_x=0$ and therefore $f'=0$. A <holomorphic function> with zero <derivative> on a connected <open set> is constant. Hence $f$ is constant, as in <analytic function with image in an affine real line>.
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