Let z be the complex number satisfying |z − 5| ≤ 3 and having maximum positive principal argument.

Then $34 {|| \frac{5 z - 12}{5 i z + 16} ||}^{2}$ is equal to: [2026]

Question

Let z be the complex number satisfying |z − 5| ≤ 3 and having maximum positive principal argument.

Then $34 {|| \frac{5 z - 12}{5 i z + 16} ||}^{2}$ is equal to: [2026]

Smart Achievers · Accepted Answer

(3)

$| z - 5 |$ $\leq 3$

For $\arg (z)$ to be maximum, $z$ lies at $P$ .

$z \equiv (4 \cos θ, 4 \sin θ)$

$\equiv (4 \cdot \frac{4}{5}, 4 \cdot \frac{3}{5}) = (\frac{16}{5}, \frac{12}{5}) = \frac{16}{5} + \frac{12 i}{5}$

Now, $34 {|| \frac{5 z - 12}{5 i z + 16} ||}^{2} = 34 {|| \frac{(16 + 12 i) - 12}{(16 i - 12) + 16} ||}^{2}$

$= 34 {|| \frac{4 + 12 i}{16 i + 4} ||}^{2}$

$= 34 (\frac{16 + 144}{256 + 16})$

$= 34 (\frac{160}{272}) = 20$

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