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Let $z=x+$ iy be a point in the Argand plane. If the amplitude of $\left(\frac{z-3}{z+2 i}\right)$ is $\frac{\pi}{2}$, then the locus of $\mathrm{z}$ is
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The correct answer is:
a semicircular are containing the origin
$\operatorname{Amp}\left(\frac{Z-3}{Z+2 i}\right)=\operatorname{Amp}(Z-3)-\operatorname{Amp}(Z+2 i)=\frac{\pi}{2}$
if $\operatorname{Amp}\left(\frac{Z-Z_1}{Z-Z_2}\right)=\frac{\pi}{2}$
$\Rightarrow$ Locus of $Z$ is an arc as semicircle.
Putting $Z=0$
$$
\operatorname{Amp}(-3)-\operatorname{Amp}(2 i) \Rightarrow \pi-\frac{\pi}{2}=\frac{\pi}{2}
$$
$\therefore \quad Z$ represents semicircle arc containing origin.
if $\operatorname{Amp}\left(\frac{Z-Z_1}{Z-Z_2}\right)=\frac{\pi}{2}$
$\Rightarrow$ Locus of $Z$ is an arc as semicircle.
Putting $Z=0$
$$
\operatorname{Amp}(-3)-\operatorname{Amp}(2 i) \Rightarrow \pi-\frac{\pi}{2}=\frac{\pi}{2}
$$
$\therefore \quad Z$ represents semicircle arc containing origin.
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