1. Tardigrade
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  3. Mathematics
  4. Let z be a complex number such that |(z-2 i/z+i)|=2, z ≠-i. Then z lies on the circle of radius 2 and centre

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Q. Let $z$ be a complex number such that $\left|\frac{z-2 i}{z+i}\right|=2, z \neq-i$. Then $z$ lies on the circle of radius $2$ and centre

JEE MainJEE Main 2023Complex Numbers and Quadratic Equations

Solution:

$ ( z -2 i )(\overline{ z }+2 i )=4( z + i )(\overline{ z }- i )$
$ z \overline{ z }+4+2 i ( z -\overline{ z })=4( z \overline{ z }+1+ i (\overline{ z }- z )) $
$ 3 z \overline{ z }-6 i ( z -\overline{ z })=0$
$ x ^2+ y ^2-2 i (2 iy )=0$
$ x ^2+ y ^2+4 y =0$