Question

# If${a^2} + {b^2} = 1$then $\dfrac{{1 + b + ia}}{{1 + b - ia}} =$?

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Hint: Complex numbers are numbers of the form axis b where ‘a’ is the real part of the complex number and ‘b’ is the imaginary part of the complex number. In complex number ${i^2} = - 1$
The multiplication of a complex number and its conjugate is formulated as:
$(a +ib)$ be the complex number and its conjugate is $(a - ib)$
$\Rightarrow (a - ib)(a - ib) = a(a - ib) + ib(a - ib)$
$= {a^2} - iab + iab - {i^2}{b^2}$
${a^2} + {b^2}$

Complete step-by- step solution:
Given ${a^2} + {b^2} = 1....(1)$
We know, ${a^2} + {b^2} = (b + ia)(b - ia)$
Proof: On multiplying, we get:
$(b + ia)(b - ia) = b(b - ia) + ia(b - ia)$
$= {b^2} - bia + iab - {i^2}{a^2}$
On cancelling $- bia$ and $iab$, we get:
$= {b^2} - {i^2}{a^2}$
Substituting ${i^2} = - 1$, we get:
$\Rightarrow (b + ia)(b - ia) = {b^2} + {a^2}.........(2)$
On using equation (2) in equation(1) we have
$\Rightarrow$ $(b + ia)(b - ia) = 1$
$\Rightarrow b - ia = \dfrac{1}{{b + ia}}..............(3)$
Now we have to find the value of:
$\dfrac{{(1 + b + ia)}}{{(1 + b + ia)}}..............(4)$
Substituting $b - ia = \dfrac{1}{{b + ia}}$ from (3) in (4), we have:
$\Rightarrow \dfrac{{1 + b + ia}}{{1 + \dfrac{1}{{b + ia}}}}$
By taking LCM, we get:
$\Rightarrow \dfrac{{1 + b + ia}}{{\dfrac{{b + ia + 1}}{{b + ia}}}}$
We know, $\{ \because \dfrac{a}{{\dfrac{b}{c}}} = \dfrac{a}{b} \times c\}$
$\Rightarrow \dfrac{{1 + b + ia}}{{1 + b + ia}} \times (b + ia)$
On cancelling$1 + b + ia$, we get
$\Rightarrow b + ia$
Hence the required value is:
$\Rightarrow \dfrac{{1 + b + ia}}{{1 + b - ia}} = b + ia$

Note: Every complex number has associated with it another complex number known as its complex conjugate. You find the complex conjugate simply by changing the sign of the imaginary part of the complex number. Example: To find the complex conjugate of $4 + 7i$ we change the sign of the imaginary part. Thus, the complex conjugate of $4 + 7i$ is$4 - 7i$. We have multiplied a complex number by its conjugate and the answer is a real number. This is a very important property which applies to every complex conjugate pair of numbers.
In conjugate of a complex number only the imaginary part of that complex number changes its sign.