Let z be complex number such that [| z - i z + 2i | = 1] and |z| = [5 2].

Question

Let z be complex number such that

$$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$ and |z| = $${5 \over 2}$$.
 Then the value of |z + 3i| is :

Accepted Answer

Given $$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$

|z -- i| = |z + 2i|

(let z = x + iy)

$$ \Rightarrow $$  x2
 + (y -- 1)2
 = x2
 + (y + 2)2

$$ \Rightarrow $$ y = $$ - {1 \over 2}$$

Also given |z| = $${5 \over 2}$$

$$ \Rightarrow $$ x2
 + y2 = $${{25} \over 4}$$

$$ \Rightarrow $$ x2 = 6

$$ \therefore $$ z = $$ \pm \sqrt 6 $$ - $$ - {1 \over 2}i$$

|z + 3i| = $$\sqrt {6 + {{25} \over 4}} $$ = $${7 \over 2}$$

Solutions · 1