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Question
If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (–4, 0), find the greatest and least values of |z + 1|.
Solution
Distance of the point representing z from the centre of the circle is |z – (–4 + i0)| = |z + 4|.
According to given condition |z + 4| ≤ 3.
Now |z + 1| = |z + 4 – 3| ≤ |z + 4| + |–3| ≤ 3 + 3 = 6
Therefore, greatest value of |z + 1| is 6.
Since least value of the modulus of a complex number is zero, the least value of |z + 1| = 0.
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