Question

Solve for a and b if $ \dfrac{1}{{a + ib}} = 3 - 2i $

Answer 1

Hint: To solve this question the first thing we should do is rearranging the given equation, after that we can rationalize it to find the value of a and b by using equality of two complex numbers. A method called rationalisation enables the division of difficult numbers that are represented in Cartesian form. Because of the imaginary part of the denominator, the creation of a fraction poses difficulties. Through multiplying the numerator and denominator by the conjugate of the denominator, the denominator can be expected to be true.

Complete step-by-step answer:
Given, $ \dfrac{1}{{a + ib}} = 3 - 2i $ .
Now, rearrange the given equation.
 \[
  \dfrac{1}{{a + ib}} = 3 - 2i \\
  a + ib = \dfrac{1}{{3 - 2i}} \;
 \]
Now, rationalize the denominator.
 \[
\Rightarrow a + ib = \dfrac{1}{{3 - 2i}} \\
\Rightarrow a + ib = \dfrac{1}{{3 - 2i}} \times \dfrac{{3 + 2i}}{{3 + 2i}} \\
\Rightarrow a + ib = \dfrac{{3 + 2i}}{{\left( {3 - 2i} \right)\left( {3 + 2i} \right)}} \\
\Rightarrow a + ib = \dfrac{{3 + 2i}}{{{3^2} - {{\left( {2i} \right)}^2}}} \\
\Rightarrow a + ib = \dfrac{{3 + 2i}}{{9 - 4{i^2}}} \\
\Rightarrow a + ib = \dfrac{{3 + 2i}}{{9 - \left( 4 \right)\left( { - 1} \right)}} \\
\Rightarrow a + ib = \dfrac{{3 + 2i}}{{9 + 4}} \\
\Rightarrow a + ib = \dfrac{{3 + 2i}}{{13}} \\
\Rightarrow a + ib = \dfrac{3}{{13}} + \dfrac{2}{{13}}i \;
 \]
Now, equating the real and imaginary parts of both sides in the above equation, we get,
 $ a = \dfrac{3}{{13}} $ and $ b = \dfrac{2}{{13}} $ .
So, the required values of a is $ \dfrac{3}{{13}} $ and b is $ \dfrac{2}{{13}} $ .
So, the correct answer is “ $ \dfrac{2}{{13}} $ ”.

Note: Every number such as positive, negative, zero, integer, rational, irrational, fractions, etc. that is found in a number system is real numbers. It is depicted as Re().
The numbers which are not real are numbers that are imaginary. It gives a negative result when we square an imaginary number.
The numbers represented in the form of \[a + ib\] where i is an imaginary number called iota and has the value of $ \sqrt { - 1} $ are complex numbers. For instance, a complex number is \[2 + 3i\] , where 2 is a real number and an imaginary number is 3i. The combination of both the true number and the imaginary number is a complex number.