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June 2021 p32 q2
1940
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z + 1 - i| \leq 1\) and \(\arg(z - 1) \leq \frac{3}{4}\pi\).
Solution
The inequality \(|z + 1 - i| \leq 1\) represents a circle centered at \(-1 + i\) with radius 1. This is because the modulus \(|z + 1 - i|\) is the distance from \(z\) to the point \(-1 + i\).
The inequality \(\arg(z - 1) \leq \frac{3}{4}\pi\) represents the region below the line that makes an angle of \(\frac{3}{4}\pi\) with the positive real axis, starting from the point \(1\).
To solve the problem, draw the circle centered at \(-1 + i\) with radius 1 on the Argand diagram. Then, draw the line starting from \(1\) with an angle of \(\frac{3}{4}\pi\) to the positive real axis. The solution is the region inside the circle and below this line.
