Question

If $arg(z-\alpha )=\frac{\pi }{4}$, where $\alpha \in R$, then the locus of $z\in Z$ is a

• A. hyperbola
• B. parabola
• C. ellipse
• D. straight line

Answer 1

The correct option is D straight line

Given, $arg(z-\alpha )=\frac{\pi }{4}$

$\Rightarrow arg\left\{\right(x-\alpha )+iy\}=\frac{\pi }{4}$

$\Rightarrow {\tan }^{-1}\left(\frac{y}{x-\alpha }\right)=\frac{\pi }{4}$

$\Rightarrow \frac{y}{x-\alpha }=\tan \frac{\pi }{4}=1$

$\Rightarrow x-\alpha =y$