The sum of the square of the modulus of the elements in the set { z = a + i b : a , b ? Z , z ? C , | z - 1 | ? 1 , | z - 5 | ? | z - 5 i | } is ____________. [2024]

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Solution

Q.
The sum of the square of the modulus of the elements in the set ${z = a + i b : a, b \in Z, z \in C, | z - 1 | \leq 1, | z - 5 | \leq | z - 5 i |}$ is ____________. [2024]

Ans.
(9)

$We have, | z - 1 | \leq 1$

$\Rightarrow {(a - 1)}^{2} + b^{2} \leq 1$

$and | z - 5 | \leq | z - 5 i |$

$\Rightarrow {(a - 5)}^{2} + b^{2} \leq {(a)}^{2} + {(b - 5)}^{2}$

$\Rightarrow a \geq b$

$z = 0 + 0 i, 1 + 0 i, 2 + 0 i, 1 + i, 1 - i$

$\Rightarrow | z | = 0, 1, 2, \sqrt{2}, \sqrt{2}$

$∴$ $Sum of | z |^{2} = 0 + 1 + 4 + 2 + 2 = 9$

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