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Q)

The locus of $z$ satisfying $\;arg(z) = \large\frac{\pi}{3}\;$ is

$(a)\;\sqrt{3} x\qquad(b)\;\sqrt{2} x\qquad(c)\;2x\qquad(d)\;x$

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Answer : $\;\sqrt{3} x$
Explanation :
Let $\;z=x+iy\;$ then its polar form is $\;z=r(cos \theta + i sin \theta)$
Where , $\; tan \theta= \large\frac{y}{x}\;$ and $\theta \;$ is $\;arg(z)\;. \theta = \large\frac{\pi}{3}\;$ thus
$ tan \large\frac{\pi}{3} = \large\frac{y}{x}$ => $\; y= \sqrt{3} x\;,$ where x > 0 , y > 0
Hence , locus of z is the part of $\;y= \sqrt{3}x\;$ in the first quadrant expect origin
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