Complex Numbers Part 1

3 $\left|z_1\right|$ = $4\left|z_2\right|$ ⇒ $\frac{\left|z_1\right|}{\left|z_2\right|}$ = $\frac{4}{3}$ ⇒ $\frac{\left|3z_1\right|}{\left|2z_2\right|}$ = 2 Let $\frac{3z_1}{2z_2}$ = a = 2 cos θ + 2i sin θ z = $\frac{3z_1}{2z_2}$ + $\frac{2z_2}{3z_1}$ = a + $\frac{1}{a}$ = $\frac{5}{2}$ cos θ + $\frac{3}{2}$ i sin θ Now all options are incorrect Remark There is a misprint in the problem actualproblem should be : "Let $z_1$ and $z_2$ be any non-zero complexnumber such that $3\left|z_1\right|$ = 2 $\left|z_2\right|$ If z = $\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$ , then Given 3 $\left|z_1\right|$ = $2\left|z_2\right|$ Now $\left|\frac{3z_1}{2z_2}\right|$ = 1 Let $\frac{3z_1}{2z_2}$ = a = cos θ + i sin θ z = $\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$ = a + $\frac{1}{a}$ = 2 cos θ ∴ Im (z) = 0 Now option (4) is correct