Question

The centre of circle represented by $\left|z+1\right|=2\left|z-1\right|$ in the complex plane is

• A. $(0,0)$
• B. $(\frac{-5}{3},0)$
• C. $(\frac{1}{3},0)$
• D. None of these

Answer 1

The correct option is D None of these

$\Rightarrow |z+1|=2|z-1|$

Let

$z=x+yi$

After taking magnitude and squaring on both side we get

$\Rightarrow \left|\right(x+1)+yi|=2\left|\right(x-1)+yi|\Rightarrow (x+1{)}^2+y^2=4({(x-1)}^2+y^2)\Rightarrow x^2+1+2x+y^2={4x}^2-8x+4+{4y}^2\Rightarrow {3x}^2+{3y}^2-10x+3=0\Rightarrow x^2+y^2-\frac{10x}{3}+1=0$

Finally the center of circle is

$(\frac{5}{3},0)$

Correct option is D