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The greatest +ve argument of complex number satisfying $|z-4|=Re(z)$ is :

$\begin{array}{1 1}(A)\;\large\frac{\pi}{3}&(B)\;\large\frac{2\pi}{3}\\(C)\;\large\frac{\pi}{2}&(D)\;\large\frac{\pi}{4}\end{array} $

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$z=x+iy$
$\mid x+iy-4\mid=x$
$\Rightarrow (x-4)^2+y^2=x^2$
$\Rightarrow y^2=8(x-2)$
Greatest argument will be at point on parabola where line through origin is tangent.
Equation of tangent $yy_1=4(x+x_1)-16$
$\Rightarrow 0(4t)=4(0+2t^2+2-16$
$\Rightarrow 8t^2+8-16=0$
$\Rightarrow t=\pm 1$
For +ve argument t=1
$\Rightarrow$ Tangent $\equiv y(4)=4(x+4)-16$
Slope =$\tan \theta=1$
$\Rightarrow \theta=arg(z)=\large\frac{\pi}{4}$
Hence (D) is the correct answer.
answered Apr 10, 2014 by sreemathi.v
 

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