Let $z$ be a complex number such that $\left|\frac{z-i}{z+2 i}\right|=1$ and $|z|=\frac{5}{2}$. Then the value of $|z+3 i|$ is:

JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Complex Numbers

Let

z

be a complex number such that

\frac{z - i}{z + 2 i} = 1

and

∣ z ∣ = \frac{5}{2}

. Then the value of

∣ z + 3 i ∣

is:

Answer: (D)

Writing

z = x + i y

, the condition

\frac{z - i}{z + 2 i} = 1

means

∣ z - i ∣ = ∣ z + 2 i ∣

; squaring,

x^{2} + (y - 1)^{2} = x^{2} + (y + 2)^{2}

, which gives

y = - \frac{1}{2}

. The constraint

∣ z ∣ = \frac{5}{2}

then forces

x^{2} + \frac{1}{4} = \frac{25}{4}

, so

x^{2} = 6

. Finally

∣ z + 3 i ∣ = x^{2} + (y + 3)^{2} = 6 + (\frac{5}{2})^{2} = \frac{49}{4} = \frac{7}{2}

JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Single correct option