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

Question

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 (A) (2,0) (B) (0,2) (C) (0,0) (D) (0,-2)

Tardigrade Admin · Accepted Answer

( 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

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

$(2, 0)$

$(0, 2)$

$(0, 0)$

$(0, - 2)$

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

Q. Let z be a complex number such that ∣∣​z+iz−2i​∣∣​=2,z=−i. Then z lies on the circle of radius 2 and centre

(2,0)

(0,2)

(0,0)

(0,−2)