First slide

Algebra of complex numbers

Question

$Of all complex numbers z satisfying |z - \frac{4}{z}| = 3, let α and β be the complex$ numbers having the greatest and the least possible modulus respectively. $Denote by Re (z) and Im (z) the real and imaginary parts respectively of the$ $complex number z .$ Which of the following statement(s) is/are false?

Difficult

A

$| α | equals 4 + 2 \sqrt{2}$
B

$| β | equals 4 - 2 \sqrt{2}$
C

$Im (α) = 0$
D

$Re (β) = 0$

Solution

$\sin c e ||z_{1}| - |z_{2}|| \leq |z_{1} - z_{2}| \Rightarrow ||z| - \frac{4}{|z|}| \leq |z - \frac{4}{z}| = 3$

$\Rightarrow |r - \frac{4}{r}| \leq 3, w h e r e r = |z| \Rightarrow - 3 \leq r - \frac{4}{r} \leq 3$
$\Rightarrow r^{2} + 3 r - 4 \geq 0 a n d r^{2} - 3 r - 4 \leq 0 \Rightarrow r \geq 1 a n d - 1 \leq r \leq 4 \Rightarrow t h e r e 1 \leq r \leq 4 \Rightarrow r = |z| m a x i m u m v a l u e = 4 a n d |z| m i n i m u m v a l u e = 1$
$a n d z = 4, z = 1 s a t i s f i e s t h e g i v e n e q u a t i o n w h o s e i m a g i n a r y p a r t i s 0 h e n c e o p t i o n s 1, 2, 4 a r e w r o n g$

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