Answer: the correct option is
(D) The imaginary part is zero.
Step-by-step explanation: Given that neither a nor b are equal to zero.
We are to select the correct statement that accurately describes the following product :
[tex]P=(a+bi)(a-bi)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We will be using the following formula :
[tex](x+y)(x-y)=x^2-y^2.[/tex]
From product (i), we get
[tex]P\\\\=(a+bi)(a-bi)\\\\=a^2-(bi)^2\\\\=a^2-b^2i^2\\\\=a^2-b^2\times (-1)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }i^2=-1]\\\\=a^2+b^2.[/tex]
So, there is no imaginary part in the given product.
Thus, the correct option is
(D) The imaginary part is zero.
