Let S = { z : | z - 6 i z - 2 i | = 1 and | z - 8 + 2 i z + 2 i | = 3 5 } . Then z S | z | 2 is equal to [2026]

Question

Let $S = {z \in ℂ : | \frac{z - 6 i}{z - 2 i} | = 1 and | \frac{z - 8 + 2 i}{z + 2 i} | = \frac{3}{5}} .$

Then $\sum_{z \in S} | z |^{2}$ is equal to [2026]

Smart Achievers · Accepted Answer

(2)

$Solving$ $| \frac{z - 6 i}{z - 2 i} |$ $= 1 \Rightarrow y = 4 \dots (1)$

$(where z = x + i y)$

$Now solving$ $| \frac{z - 8 + 2 i}{z + 2 i} |$ $= \frac{3}{5}$

$\Rightarrow x^{2} + y^{2} - 25 x + 4 y + 104 = 0 \dots (2)$

$Solving (1) & (2) \Rightarrow z = 17 + 4 i$ $&$ $8 + 4 i$

$\Rightarrow \sum | z |^{2} = {(17)}^{2} + {(4)}^{2} + {(8)}^{2} + {(4)}^{2} = 385$

Solution

Q.
Let $S = {z \in ℂ : | \frac{z - 6 i}{z - 2 i} | = 1 and | \frac{z - 8 + 2 i}{z + 2 i} | = \frac{3}{5}} .$

Then $\sum_{z \in S} | z |^{2}$ is equal to [2026]

1 423

2 385

3 413

4 398

Want Us to Email you About Special Offers & Updates?