Complex NumberseasyPYQ · JEE Main · 4 Apr 2026 · Shift 2 (Afternoon)Free

Sum of |z+√3 i|^2 for Roots of z^2+4z+16: 38 | JEE 2026

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

Question
Let S={z∈C:z2+4z+16=0}S=\{z\in\mathbb{C}:z^2+4z+16=0\}. Then ∑z∈S∣z+3 i∣2\displaystyle\sum_{z\in S}\left|z+\sqrt3\,i\right|^2 is equal to
A4242
B2323
C2727
D3838correct
Solution
Step 1: Solve z2+4z+16=0z^2+4z+16=0:
(z+2)2=−12 ⇒ z=−2±23 i.(z+2)^2=-12\ \Rightarrow\ z=-2\pm2\sqrt3\,i.
 So S={−2+23 i, −2−23 i}S=\{-2+2\sqrt3\,i,\ -2-2\sqrt3\,i\}. Step 2: Compute each ∣z+3 i∣2\left|z+\sqrt3\,i\right|^2:
∣−2+23 i+3 i∣2=∣−2+33 i∣2=4+27=31,\left|-2+2\sqrt3\,i+\sqrt3\,i\right|^2=\left|-2+3\sqrt3\,i\right|^2=4+27=31,
∣−2−23 i+3 i∣2=∣−2−3 i∣2=4+3=7.\left|-2-2\sqrt3\,i+\sqrt3\,i\right|^2=\left|-2-\sqrt3\,i\right|^2=4+3=7.
 Step 3: Sum:
31+7=38.31+7=38.
 Correct answer: (4)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Complex Numbers · easy
If ∣z∣=min⁡{∣z−1∣, ∣z+1∣}|z| = \min\{|z - 1|,\, |z + 1|\}, then the value of ∣z+zˉ∣|z + \bar{z}| is
Complex Numbers · easy
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
Complex Numbers · easy
If α\alpha is a root of the equation x2−x+1=0x^{2} - x + 1 = 0, then find the value of ∣α333+α666+α999∣.\left|\alpha^{333} + \alpha^{666} + \alpha^{999}\right| .
Complex Numbers · hard
Let zk=cos⁡(2kπ10)+isin⁡(2kπ10)z_k = \cos\left(\dfrac{2k\pi}{10}\right) + i\sin\left(\dfrac{2k\pi}{10}\right); k=1,2,…,9k = 1, 2, \ldots, 9. Match each entry in List-I with the correct entry in List-II.

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath