[UPLIFTJEE]

Let $\mathrm{w}_{1}$ be the point obtained by the rotation of $\mathrm{z}_{1}=5+4 \mathrm{i}$ about the origin through a right angle in the anticlockwise direction, and $\mathrm{w}_{2}$ be the point obtained by the rotation of $\mathrm{z}_{2}=3+5 \mathrm{i}$ about the origin through a right angle in the clockwise direction. Then the principal argument of $\mathrm{w}_{1}-\mathrm{w}_{2}$ is equal to [JEE (Main) 11 April 2023 SI]
(A) $-\pi+\tan ^{-1} \frac{33}{5}$
(B) $-\pi+\tan ^{-1} \frac{8}{9}$
(C) $\pi-\tan ^{-1} \frac{8}{9}$
(D) $\pi-\tan ^{-1} \frac{33}{5}$

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[UPLIFTJEE]

Solution

$\mathrm{w}_{1}=\mathrm{z}_{\mathrm{i}} \mathrm{i}=(5+4 \mathrm{i}) \mathrm{i}=-4+5 \mathrm{i}$

\[ W_{2}=z_{2}(-i)=(3+5 i)(-i)=5-4 i \]

\[ W_{1}-W_{2}=-9+8 i . \text { Principal argument }=\pi-\tan ^{-1}\left(\frac{8}{9}\right) \]

[UPLIFTJEE]

Answer

C

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