\(If a + ib =\frac{(x+1)^2}{2x^2+1},\,prove\, that\, a^2 + b^2 =\frac{(x^2+1)^2}{(2x^2+1)}\)
Solution and Explanation
\(If a + ib =\frac{(x+1)^2}{2x^2+1}\)
\(=\frac{x^2+i^2+2xi}{2x^2+1}\)
\(=\frac{x^2-1+i2x}{2x^2+1}\)
\(=\frac{x^2-1}{2x^2+1}+1(\frac{2x}{2x^2+1})\)
on comparing real and imaginary parts, we obtain
\(a=\frac{x^2-1}{2x^2+1}\,and\,\,b=\frac{2x}{2x^2+1}\)
\(a^2+b^2=(\frac{x^2-1}{2x^2+1})+(\frac{2x}{2x^2+1})^2\)
\(=\frac{x^4+1-2x^2+4x^2}{(2x+1)^2}\)
\(\frac{x^2+1+2x^2}{(2x^2+1)^2}\)
\(=\frac{(x^2+1)^2}{(2x^2+1)^2}\)
\(=a^2+b^2=\frac{(x^2+1)^2}{(2x+1)^2}\)
Hence, proved.
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.