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The complex number $z$ satisfying the equation $arg\large(\frac{z-1}{z+1})=\frac{\pi}{4}$ represents ?

$\begin{array}{1 1}(A) \;A\;Circle \\(B)\;A\;Parabola\\(C)\;A\;Straight \;Line \\(D)\;An\;Ellipse \end{array}$

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  • $arg\large(\frac{z_1}{z_2})$$=argz_1-argz_2$
Let $z=x+iy$
Given $arg\large(\frac{z-1}{z+1})=\frac{\pi}{4}$
$\Rightarrow\:arg\large(\frac{(x-1)+iy}{(x+1)+iy})=\frac{\pi}{4}$
$\Rightarrow\:arg((x-1)+iy)-arg((x+1)+iy)=\large\frac{\pi}{4}$
$\Rightarrow\:tan^{-1}\large\frac{y}{x-1}$$-tan^{-1}\large\frac{y}{x+1}=\frac{\pi}{4}$
$\Rightarrow\:tan^{-1}\bigg(\Large\frac{\large\frac{y}{x-1}-\frac{y}{x+1}}{1+\large\frac{y}{x-1}.\frac{y}{x+1}}\bigg)$$\large=\frac{\pi}{4}$
$\Rightarrow\:\Large\frac{y(x+1)-y(x-1)}{x^2-1+y^2}$$=tan\large\frac{\pi}{4}$ $=1$
$\Rightarrow\: 2y=x^2+y^2-1$
$\Rightarrow\:x^2+y^2-2y-1=0$
which is eqn., of a circle
answered Jul 15, 2013 by rvidyagovindarajan_1
 

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