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9709 P23 - Nov 2023 - Q2
1928
On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z - 1 + 2i| \leq |z|\) and \(|z - 2| \leq 1\).
Solution
1. The inequality \(|z - 2| \leq 1\) represents a circle centered at (2, 0) with radius 1.
2. The inequality \(|z - 1 + 2i| \leq |z|\) represents the region below the perpendicular bisector of the line joining the point (1, -2) and the origin (0, 0).
3. The perpendicular bisector can be found by determining the midpoint of the line segment joining (1, -2) and (0, 0), which is \((0.5, -1)\), and finding the line perpendicular to the segment through this point.
4. The correct region to shade is the intersection of the circle and the region below the perpendicular bisector.
