If a complex number $z$ lies in the interior or on the boundary of a circle of radius $3$ and centre at $(- 4, 0)$ , then prove that the greatest and least values of $| z + 1 |$ are $6, 0$ .

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Solution

$| z + 1 | = | z + 4 - 3 | = | (z + 4) + (- 3) | \leq | z + 4 | + | - 3 | = | z + 4 | + 3 \leq 3 + 3 = 6$ \ $[∵ | z + 4 | \leq 3]$
Hence the greatest value of $| z + 1 |$ is $6$ .
$| z + 1 | = | z + 4 - 3 | \geq | z + 4 | - | 3 | = 0$
Hence the least value is $0$ .

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If a complex number z lies in the interior or on the boundary of a circle of radius 3 and centre at (−4,0), then prove that the greatest and least values of |z+1| are 6,0.

|z+1|=|z+4−3|=|(z+4)+(−3)|≤|z+4|+|−3|=|z+4|+3≤3+3=6 \ [∵|z+4|≤3]Hence the greatest value of |z+1| is 6.|z+1|=|z+4−3|≥|z+4|−|3|=0Hence the least value is 0.

If a complex number $z$ lies in the interior or on the boundary of a circle of radius $3$ and centre at $(- 4, 0)$ , then prove that the greatest and least values of $| z + 1 |$ are $6, 0$ .

$| z + 1 | = | z + 4 - 3 | = | (z + 4) + (- 3) | \leq | z + 4 | + | - 3 | = | z + 4 | + 3 \leq 3 + 3 = 6$ \ $[∵ | z + 4 | \leq 3]$
Hence the greatest value of $| z + 1 |$ is $6$ .
$| z + 1 | = | z + 4 - 3 | \geq | z + 4 | - | 3 | = 0$
Hence the least value is $0$ .