Question

For the complex number z, which of the following is true?

• A. $z+\overline{z}$ is real and $z\overline{z}$ is imaginary
• B. $z+\overline{z}$ is imaginary and $z\overline{z}$ is real
• C. Both $z+\overline{z}$ and $z\overline{z}$ are real.
• D. Both $z+\overline{z}$ and $z\overline{z}$ are imaginary numbers

Answer 1

The correct option is C

Both $z+\overline{z}$ and $z\overline{z}$ are real.

Let $z=x+iy$

Here, $z+\overline{z}=(x+iy)+(x-iy)=2x$ (Real)

And $z\overline{z}=(x+iy)(x-iy)=x^2+y^2$ (Real).