Question

The locus of z satisfying arg(z) = ${\displaystyle \frac{\pi }{3}}$ is ______.

Answer 1

The locus of z satisfying arg (z) = ${\displaystyle \frac{\pi }{3}}$ is ${\displaystyle \underline{y=\sqrt{3}x}}$.

Explanation:

Let z = x + iy,

Then its polar form is z = r(cosθ + isinθ), Where tanθ = ${\displaystyle \frac{y}{x}}$ and θ is arg(z).

Given that θ = ${\displaystyle \frac{\pi }{3}}$.

Thus, ${\displaystyle \tan \frac{\pi }{3}=\frac{y}{x}}$ ⇒ y = ${\displaystyle \sqrt{3}x}$, Where x > 0, y > 0.

Hence, locus of z is the part of y = ${\displaystyle \sqrt{3}x}$ in the first quadrant except origin.