Question:
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are _______ and ____________.
Solution:
Given |z + 4| ≤ 3
here |z + 1| = |z + 1 + 3 – 3| = |z + 4 + (–3)|
Since |a + b| ≤ |a| + |b| ≤ |z + 4| + |–3| = |z + 4| + 3
≤ 3 + 3 (given)
hence maximum value of |z + 1| is 6
|z + 1| = |z + 4 –3|
Since |a – b| ≥ ||a| – |b|| ≥ –|a| + |b|
⇒ |z + 1| ≥ – |z + 4| + 3
Since |z + 4| ≤ 3
⇒ –|z + 4| ≥ –3
i.e |z + 1| ≥ –|z + 4| + 3 ≥ –3 + 3 = 0
Hence, minimum value of |z + 1| is 0.
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