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Evaluate |4z2z3+9z3z1+16z1z2| with Given Moduli | JEE

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

Question
If ∣z1∣=2|z_1| = 2, ∣z2∣=3|z_2| = 3, ∣z3∣=4|z_3| = 4 and ∣z1+z2+z3∣=5|z_1 + z_2 + z_3| = 5, then ∣4z2z3+9z3z1+16z1z2∣|4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2| equals
A2020
B2424
C4848
D120120correct
Solution
Step 1: Convert the given moduli into conjugate relations. For any complex number, ∣z∣2=zzˉ|z|^2 = z\bar{z}, so zˉ=∣z∣2z\bar{z} = \dfrac{|z|^2}{z}. Hence:
∣z1∣2=4  ⟹  z1ˉ=4z1,∣z2∣2=9  ⟹  z2ˉ=9z2,∣z3∣2=16  ⟹  z3ˉ=16z3|z_1|^2 = 4 \implies \bar{z_1} = \frac{4}{z_1}, \quad |z_2|^2 = 9 \implies \bar{z_2} = \frac{9}{z_2}, \quad |z_3|^2 = 16 \implies \bar{z_3} = \frac{16}{z_3}
 Step 2: Factor z1z2z3z_1 z_2 z_3 out of the target expression:
4z2z3+9z3z1+16z1z2=z1z2z3(4z1+9z2+16z3)4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2 = z_1 z_2 z_3\left(\frac{4}{z_1} + \frac{9}{z_2} + \frac{16}{z_3}\right)
 Step 3: Replace each fraction using Step 1:
4z1=z1ˉ,9z2=z2ˉ,16z3=z3ˉ\frac{4}{z_1} = \bar{z_1}, \quad \frac{9}{z_2} = \bar{z_2}, \quad \frac{16}{z_3} = \bar{z_3}
 So:
4z2z3+9z3z1+16z1z2=z1z2z3 (z1ˉ+z2ˉ+z3ˉ)=z1z2z3 (z1+z2+z3)‾4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2 = z_1 z_2 z_3\,(\bar{z_1} + \bar{z_2} + \bar{z_3}) = z_1 z_2 z_3\,\overline{(z_1 + z_2 + z_3)}
 Step 4: Take the modulus. Using ∣wv∣=∣w∣∣v∣|wv| = |w||v| and ∣uˉ∣=∣u∣|\bar{u}| = |u|:
∣4z2z3+9z3z1+16z1z2∣=∣z1∣ ∣z2∣ ∣z3∣ ∣z1+z2+z3∣|4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2| = |z_1|\,|z_2|\,|z_3|\,|z_1 + z_2 + z_3|
 Step 5: Substitute the values:
=2⋅3⋅4⋅5=120= 2 \cdot 3 \cdot 4 \cdot 5 = 120
 Correct answer: (4)
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