The largest value of r for which the region represented by the set {ω∈C:∣ω−4−i∣≤r} is contained in the region represented by the set {z∈C:∣z−1∣≤∣z+i∣} is equal to:
Solutions · 1
Official solution
∣z−1∣≤∣z+i∣ means z is at least as close to (1,0) as to (0,−1), i.e.
\[(x-1)^2+y^2\le x^2+(y+1)^2\ \Rightarrow\ x+y\ge0.\]
The disc ∣ω−4−i∣≤r is centred at (4,1). The largest such disc contained in the half-plane x+y≥0 has radius equal to the perpendicular distance from (4,1) to the line x+y=0:
\[r=\frac{|4+1|}{\sqrt{2}}=\frac{5}{\sqrt{2}}.\]
*(The option's exact fraction layout was corrupted in the OCR; r=5/2 is the value consistent with the geometry.)*