JEE PYQ: Complex Numbers Question 38
Question 38 - 2019 (10 Jan Shift 1)
Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4|z_2|$. If $z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$ then:
(1) $3/2 \le |z| \le 5/2$
(2) $1/2 \le |z| \le 5/2$
(3) $3/2 \le |z| \le 7/2$
(4) $1 \le |z| \le 5/2$
Show Answer
Answer: (1)
Solution
Let $w = \frac{3z_1}{2z_2}$. Then $|w| = \frac{3|z_1|}{2|z_2|} = \frac{4|z_2|}{2|z_2|} = 2$. $z = w + \frac{1}{w}$. $|z| = |w + 1/w|$. With $|w| = 2$: $||w| - |1/w|| \le |z| \le |w| + |1/w|$, i.e., $\frac{3}{2} \le |z| \le \frac{5}{2}$.
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