Question

Assertion :The locus of the point $z$ satisfying the condition $arg\frac{z-1}{z+1}=\frac{\pi }{3}$ is a parabola Reason: $arg\frac{z_1}{z_2}=arg{z}_1-arg{z}_2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is D Both Assertion and Reason are incorrect

We have, $arg\frac{z-1}{z+1}=\frac{\pi }{3}\Rightarrow arg\frac{x+iy-1}{x+iy+1}=\frac{\pi }{3}$ $[$ putting $z=x+iy]$

$\Rightarrow {\tan }^{-1}\frac{y}{x-1}-{\tan }^{-1}\frac{y}{x+1}=\frac{\pi }{3}(\because arg\frac{z_1}{z_2}=arg{z}_1-arg{z}_2)$

$\Rightarrow {\tan }^{-1}\frac{\frac{y}{x-1}-\frac{y}{x+1}}{1+\frac{y^2}{x^2-1}}=\frac{\pi }{3}\Rightarrow \frac{2y}{x^2+y^2-1}=\tan \frac{\pi }{3}=\sqrt{3}$

$\Rightarrow x^2+y^2-\frac{2}{\sqrt{3}}y-1=0,$ which is a circle.

$arg\frac{z_1}{z_2}=arg{z}_1-arg{z}_2+2\pi n$ .... n=0 if $-\pi /2<arg{z}_1-arg{z}_2<\pi /2$