Question

1\a+bi=3-2i find and b
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Answer 1

Answer:  The required values of a and b are

$a=\dfrac{3}{13},~~b=\dfrac{2}{13}.$

Step-by-step explanation:  We are given the following equality involving complex numbers :

$\dfrac{1}{a+bi}=3-2i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We are to find the value of a and b.

From equation (i), we have

$\dfrac{1}{a+bi}=3-2i\\\\\\\Rightarrow a+bi=\dfrac{1}{3-2i}\\\\\\\Rightarrow a+bi=\dfrac{3+2i}{(3-2i)(3+2i)}\\\\\\\Rightarrow a+bi=\dfrac{3+2i}{3^2-(2i)^2}\\\\\\\Rightarrow a+bi=\dfrac{3+2i}{9+4}~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }i^2=-1]\\\\\\\Rightarrow a+bi=\dfrac{3+2i}{13}\\\\\\\Rightarrow a+bi=\dfrac{3}{13}+\dfrac{2}{13}i.$

Equating the real and imaginary parts of both sides, we get

$a=\dfrac{3}{13},~~b=\dfrac{2}{13}.$

Thus, the required values of a and b are

$a=\dfrac{3}{13},~~b=\dfrac{2}{13}.$