If $| z - 5 - 7 i | = 9,$ then find the greatest and least values of $| z - 2 - 3 i | .$

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Solution

We have $9 = | z - (5 + 7 i) | =$ distance between $z$ and $5 + 7 i .$
Thus locus of $z$ is the circle of radius $9$ and centre at $5 + 7 i .$ For such a $z$ (on the circle), we have to find its greatest and least distance as from $2 + 3 i,$ which obviously $14$ and $4.$

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