Equations involving complex number z and its conjugate

Prove that: (i) the sum of a complex number and its conjugate is real (ii) the product of a complex number and its conjugate is real.

Statement I: We have an equation involving the complex number z is |(z-3i)/(z+3i)| which lies on the x-axis.Statement II: The equation of the x -axis is y=3

Find the sum and product of the complex number (-1 +2i) with its conjugate.

Find the sum and product of the complex number (3-4i) with its conjugate.

Complex Number Equation Solving | Properties of Modulus and Conjugate

Problems based on equations involving Complex numbers

If a complex number Z is purely imaginary then conjugate of Z is

If a complex number Z is purely imaginary then conjugate of Z is