Question

The product of a complex number z = x + iy and its complex conjugate z̅ is
1. x²
2. y²
3. x² – y²
4. x² + y²

Answer 1

Correct Answer - Option 4 : x² + y²

Concept:

If, z = x + iy is a complex number, then

Conjugate of z = z̅ = x - iy  

∴ The product of a complex number z = x + iy and its complex conjugate z̅  is zz̅ = (x + iy)(x - iy)

Calculation:

Given:

z = x + iy

∴ zz̅ = (x + iy)(x - iy)

 zz̅  = \(x^2 - xyi + xyi + y^2\)      [∵ i2 = -1]

 zz̅ = \(x^2 + y^2\)

Hence, option (4) is correct answer.

Modulus of z:

\(\left| {\rm{z}} \right| = \sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = \sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{\;Im\;}}{{\left( {\rm{z}} \right)}^2}} \)