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If a2+b2=1 then 1+b+ia1+bia is equal to
A. 1
B. 2
C. b+ia
D. a+ib

Answer
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Hint: A complex number is of the form a+ib where a and b are real numbers and i is an imaginary number. We will use the formula (ab)(a+b)=a2b2 and (a+b)2=a2+b2+2ab to solve the given expression. Also, we know that i=1. Here, first we need to multiply by 1+b+ia and then, we will make the given expression in the form of a and b after.

Complete step by step answer:
We need to find the value of the given expression as below,
1+b+ia1+bia
Multiply the numerator and denominator by 1+b+ia, we will get,
1+b+ia1+bia=(1+b)+(ia)(1+b)(ia)×(1+b)+(ia)(1+b)+(ia)
We have used brackets so that we can takea=1+b and b=ia(this is an assumption).
We will use the formula (ab)(a+b)=a2b2 and applying in the expression, we get,
1+b+ia1+bia=[(1+b)+(ia)]2(1+b)2(ia)2
We will use the formula(a+b)2=a2+b2+2ab and applying it in the expression, we get,
1+b+ia1+bia=(1+b)2+(ia)2+2(1+b)(ia)(1+b2+2b)i2a2

Again we will use the formula (a+b)2=a2+b2+2ab and applying it in the expression, we get,
1+b+ia1+bia=(1+b2+2b)+i2a2+2i(a+ab)1+b2+2bi2a2
We know thati=1i2=1.
Substituting the value ofiin above expression, we will get,
1+b+ia1+bia=(1+b2+2b)+(1)a2+2i(a+ab)1+b2+2b(1)a2
Removing the brackets, we get,
1+b+ia1+bia=1+b2+2ba2+2ia(1+b)1+b2+2b+a2
We are given that a2+b2=1.

So, here we will substitute 1=a2+b2 in numerator and a2+b2=1 in denominator, we will get,
1+b+ia1+bia=a2+b2+b2+2ba2+2ia(1+b)1+1+2b2
Simplify this above expression, we get,
1+b+ia1+bia=2b2+2b+2ia(1+b)2+2b
1+b+ia1+bia=2b(b+1)+2ia(1+b)2(1+b)
Dividing by 2in both numerator and denominator, we get,
1+b+ia1+bia=b(b+1)+ia(1+b)(1+b)
Rearrange this above expression, we get,
1+b+ia1+bia=b(1+b)+ia(1+b)(1+b)
1+b+ia1+bia=b(1+b)(1+b)+ia(1+b)(1+b)
Dividing by(1+b)in both numerator and denominator, we get,
1+b+ia1+bia=b+ia

Hence, if a2+b2=1 then 1+b+ia1+bia is b+ib.

Note: Complex Number is an algebraic expression including the factor i=1. These numbers have two parts, one is called the real part and is denoted by Re (z) and the other is called the Imaginary Part called “iota”. Imaginary part is denoted by Im (z) for the complex number represented by 'z'. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.