First slide

Algebra of complex numbers

Question

If z = (3 + 7i) (a + ib) where $a, b \in Z - {0}$ , is purely imaginary, then the minimum value of |z|² is

Moderate

A

74
B

45
C

58
D

65

Solution

$z = (3 a - 7 b) + i (3 b + 7 a)$
For z to be purely imaginary we have
$\begin{array}{l} 3 a = 7 b \\ \Rightarrow a = 7 or b = 3 (for least value of | z |) \\ \Rightarrow | z | = \sqrt{9 + 49} = \sqrt{58} \end{array}$

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