Complex NumberseasyPYQ · JEE Main · 4 Apr 2026 · Shift 1 (Morning)Free

Complex Number with Argument Condition: |z|² = 9 | JEE 2026

JEE Maths question with a full step-by-step solution.

Question
Let zz be a complex number such that ∣z+2∣=∣z−2∣|z+2|=|z-2| and arg⁡(z+3z−i)=π4\arg\left(\dfrac{z+3}{z-i}\right)=\dfrac{\pi}{4}. Then ∣z∣2|z|^2 is equal to
A99correct
B44
C55
D11
Solution
Step 1: ∣z+2∣=∣z−2∣|z+2|=|z-2| means zz is equidistant from 22 and −2-2, so zz lies on the imaginary axis:
z=0+iy,y∈R.z=0+iy,\quad y\in\mathbb{R}.
 Step 2: Then
arg⁡(3+iyiy−i)=arg⁡(yy−1+3i1−y)=π4.\arg\left(\frac{3+iy}{iy-i}\right)=\arg\left(\frac{y}{y-1}+\frac{3i}{1-y}\right)=\frac{\pi}{4}.
 Step 3: For argument π4\dfrac{\pi}{4}, the real and imaginary parts are equal and positive:
yy−1=31−y>0 ⇒ y=−3.\frac{y}{y-1}=\frac{3}{1-y}>0\ \Rightarrow\ y=-3.
 So z=−3iz=-3i. Step 4:
∣z∣2=02+(−3)2=9.|z|^2=0^2+(-3)^2=9.
 Correct answer: (1)
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