Question

(Franciscanas-SP) O valor de a, a ∈ aos reais, que torna real o quociente $\frac{3 - 2ai}{4-3i}$ é:

a) -3/2
b) -9/8
c) 0
d) 9/8

Answer 1

$\frac{3-2ai}{4-3i}\times\frac{4+3i}{4+3i}=\\\\\\\frac{12+9i-8ai-4ai^2}{16-9i^2}=\\\\\\\frac{12+9i-8ai+4a}{16+9}=\\\\\\\frac{12+4a+9i-8ai}{25}=\\\\\\\frac{12+4a}{25}+\frac{(9-8a)i}{25}$
 
 Como podes notar, o quociente será real se a parte imaginário for nula!
 
 Por conseguinte,

$\frac{(9-8a)}{25}=0\\\\9-8a=0\\-8a=-9\\\boxed{a=\frac{9}{8}}$