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What is the locus of z , if amplitude of $\;z-2-3i\;$ is $\;\large\frac{\pi}{4}\;$ ?

(a) Straight line\[\](b) Parabola\[\](c) Not a straight line\[\](d) Semi parabola
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Answer : Straight line
Explanation :
Let $\; z=x+iy$
Then , $\;z-2-3i\; = (x-2)+i(y-3)$
Let $\;\theta\;$ be the amplitude of $\;z-2-3i\;$
Then , $\;tan \theta = \large\frac{y-3}{x-2}$
$\;tan \large\frac{\pi}{4} = \large\frac{y-3}{x-2}\qquad [\theta= \large\frac{\pi}{4}]$
$\;1 = \large\frac{y-3}{x-2}$
i.e. $\;x-y+1=0$
Hence , the locus of z is a straight line .
answered Apr 18, 2014 by yamini.v

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