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June 2023 p32 q3
1906
(a) On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(|z + 3 - 2i| = 2\).
(b) Find the least value of \(|z|\) for points on this locus, giving your answer in an exact form.
Solution
(a) The equation \(|z + 3 - 2i| = 2\) represents a circle on the Argand diagram with center at \((-3, 2)\) and radius 2. To sketch this, plot the center at \((-3, 2)\) and draw a circle with radius 2.
(b) To find the least value of \(|z|\), calculate the distance from the origin \((0, 0)\) to the center \((-3, 2)\), which is \(\sqrt{(-3)^2 + 2^2} = \sqrt{13}\). The least value of \(|z|\) is the distance from the origin to the nearest point on the circle, which is \(\sqrt{13} - 2\).