Correct Answer - Option 4 : a = b
Concept:
A complex number (Z): Complex number is the combination of a real number and an imaginary number. It is given by
Z = x + iy, where 'x' and 'y' are the real and imaginary part of Z and i = √-1
Conjugate of a complex number: When the i of a complex number is replaced with - i, we get the conjugate of that complex number.
\(\bar{Z}\ =\ x\ -\ iy\)
Re(Z) = x
\Img(Z) = y
|Z| = \(\sqrt{x^2\ +\ y^2}\)
Formula used:
1. \( |\frac{Z_1}{Z_2}| = \frac{|Z_1|}{|Z_2|}\)
2. \(Z\bar{Z}\ = |Z|^2\)
3. (a - b)2 = a2 + b2 - 2ab
Calculation:
Given that,
\(\sqrt{2ab}(\frac{Z}{\bar{Z}})\ =\ a + ib\) -----(1)
Z = (b + ia) ----(2)
Therefore, a conjugate of Z
Z̅ = b - ia ----(3)
Hence, from equation (1)
\(\sqrt{2ab}(\frac{b\ +\ ia}{b\ -\ ia})\ =\ a + ib\)
Taking modulus of both sides,
\(\sqrt{2ab}|(\frac{b\ +\ ia}{b\ -\ ia})|\ =\ |a + ib|\)
\(\sqrt{2ab}\frac{|b\ +\ ia|}{|b\ -\ ia|}|\ =\ |a + ib|\)
\(\sqrt{2ab}\frac{\sqrt{b^2\ +\ a^2}}{\sqrt{b^2\ +\ a^2}}\ = \sqrt{a^2\ +\ b^2}\)
Taking square of both side
a2 + b2 - 2ab = 0
⇒ (a - b)2 = 0
⇒ a = b
Hence, option 4 is correct.