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.