Let , and . Then is equal to __________. [2025]

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Solution

Q.
Let $A = {z \in C : | z - 2 - i | = 3}$ , $B = {z \in C : R e (z - i z) = 2}$ and $S = A \cap B$ . Then $\sum_{z \in S} | z |^{2}$ is equal to __________. [2025]

Ans.
(22)

Let z = x + iy.

We have, A : |z – 2 – i| = 3

$\Rightarrow$ $| (x - 2) + i (y - 1) |$ $= 3 \Rightarrow {(x - 2)}^{2} + {(y - 1)}^{2} = 9$ ... (i)

Also, B : Re(z – iz) = 2

$R e ((x + y) + i (y - x)) = 2 \Rightarrow x + y = 2$ ... (ii)

From (i) and (ii), we get $x = \frac{3 \pm \sqrt{17}}{2}, y = \frac{1 \mp \sqrt{17}}{2}$

$∴ \sum_{z \in S} | z |^{2} = \frac{1}{4} [2 \times 26 + 2 \times 18] = 22$ .

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