The number of complex numbers z such that |z - 1| = |z + 1| = |z - i| equals

Question

The number of complex numbers z such that |z - 1| = |z + 1| = |z - i| equals (A) 1 (B) 2 (C) ∞ (D) 0

Tardigrade Admin · Accepted Answer

Let z = x + iy |z - 1| = |z + 1| ⇒ Re z = 0 ⇒ x = 0 |z - 1| = |z - i| ⇒ x = y |z + 1| = |z - i| ⇒ y = -x Only (0, 0) will satisfy all conditions. ⇒ Number of complex number z = 1

Q. The number of complex numbers $z$ such that $∣ z - 1∣ = ∣ z + 1∣ = ∣ z - i ∣$ equals

$1$

$2$

$\infty$

$0$

Let $z = x + i y$
$∣ z - 1∣ = ∣ z + 1∣ \Rightarrow R e z = 0 \Rightarrow x = 0$
$∣ z - 1∣ = ∣ z - i ∣ \Rightarrow x = y$
$∣ z + 1∣ = ∣ z - i ∣ \Rightarrow y = - x$
Only $(0, 0)$ will satisfy all conditions.
$\Rightarrow$ Number of complex number $z = 1$

Q. The number of complex numbers z such that ∣z−1∣=∣z+1∣=∣z−i∣ equals

1

2

∞

0

Let z=x+iy ∣z−1∣=∣z+1∣⇒Rez=0⇒x=0 ∣z−1∣=∣z−i∣⇒x=y ∣z+1∣=∣z−i∣⇒y=−x Only (0,0) will satisfy all conditions. ⇒ Number of complex number z=1

Q. The number of complex numbers $z$ such that $∣ z - 1∣ = ∣ z + 1∣ = ∣ z - i ∣$ equals

$1$

$2$

$\infty$

$0$

Let $z = x + i y$
$∣ z - 1∣ = ∣ z + 1∣ \Rightarrow R e z = 0 \Rightarrow x = 0$
$∣ z - 1∣ = ∣ z - i ∣ \Rightarrow x = y$
$∣ z + 1∣ = ∣ z - i ∣ \Rightarrow y = - x$
Only $(0, 0)$ will satisfy all conditions.
$\Rightarrow$ Number of complex number $z = 1$