Highly Complex!

Algebra Level 3

If z 1 ≠ 0 , z 2 \large \displaystyle z_1 \not= 0, z_2 be two complex number such that z 2 z 1 \large \displaystyle \frac{z_2}{z_1} is pure imaginary.

Then find the value of ∣ 5 z 1 + 7 z 2 5 z 1 − 7 z 2 ∣ . \large \displaystyle \left \vert \frac{5z_1 + 7z_2}{5z_1 - 7z_2}\right \vert .


The answer is 1.00.

Samara Simha Reddy by Samara Simha Reddy , 22, India

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1 solution

Let z 2 z 1 = k i , where k ≠ 0 a real number. \text{Let} \frac{z_2}{z_1} = k_i , \text{where } k \not= 0 \text{ a real number.}

∣ 5 z 1 + 7 z 2 5 z 1 − 7 z 2 ∣ = ∣ 5 + 7 k i 5 − 7 k i ∣ = 1 \large \displaystyle \left \vert \frac{5z_1 + 7z_2}{5z_1 - 7z_2}\right \vert = \left \vert \frac{5 + 7k_i}{5 - 7k_i}\right \vert = 1 ( ∵ 5 + 7 k i and 5 − 7 k i are conjugate. ) \left(\because 5+7k_i \text{ and } 5-7k_i \text{ are conjugate.}\right)

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