If |Z|= 1 , then the locus of Z1 which moves such thatZ1=5−4Z4Z−2in the complex plane, is

z1=5−4z4z−2⇒z=5+2z14z1+1|z|=1⇒5+2z14z1+1=1 ⇒z1+52=2z1+1⇒x+522+y2=4(x+1)2+y2x2+y2+x−34=0⇒ centre =−12,0 radius =1 circle passes through all 4 quadrants

If |Z|= 1 , then the locus of Z1 which moves such thatZ1=5−4Z4Z−2in the complex plane, is

z1=5−4z4z−2⇒z=5+2z14z1+1|z|=1⇒5+2z14z1+1=1 ⇒z1+52=2z1+1⇒x+522+y2=4(x+1)2+y2x2+y2+x−34=0⇒ centre =−12,0 radius =1 circle passes through all 4 quadrants

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If |Z|= 1 , then the locus of Z₁ which moves such that $Z_{1} = \frac{5 - 4 Z}{4 Z - 2}$ in the complex plane, is

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circle centred at $(\frac{- 1}{2}, 0)$ and of radius 1

circle passing through all quadrants

circle centred at $(\frac{1}{2}, 0)$

circle of radius $4 \frac{1}{2}$

answer is C, D.

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Detailed Solution

$z_{1} = \frac{5 - 4 z}{4 z - 2}$

$\Rightarrow z = \frac{5 + 2 z_{1}}{4 (z_{1} + 1)} | z | = 1 \Rightarrow |\frac{5 + 2 z_{1}}{4 (z_{1} + 1)}| = 1$

$\begin{array}{l} \Rightarrow |z_{1} + \frac{5}{2}| = 2 |z_{1} + 1| \\ \Rightarrow {(x + \frac{5}{2})}^{2} + y^{2} = 4 ((x + 1)^{2} + y^{2}) \\ x^{2} + y^{2} + x - \frac{3}{4} = 0 \Rightarrow centre = (- \frac{1}{2}, 0) radius = 1 \end{array}$