The map is $w=(3-z)/(2z-1)$. A circle maps to a line iff it contains the pole 1/2, so k=1/2. Invert to z=(w+3)/(2w+1); |z−1|=1/2 becomes 4|w−2|²=|2w+1|², simplifying to 4x−3=0. Choose α=4,β=0,γ=−3. Required ratio=(4−3)/(4+3)=**1/7, A**.
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