Question

If neither a nor b are equal to zero, which answer most accurately describes the product of (a + bi)(a - bi)?

The real part is not positive.
The imaginary part is positive.
The real part is zero.
The imaginary part is zero.

Answer 1

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 :

`P=(a+bi)(a-bi)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

We will be using the following formula :

`(x+y)(x-y)=x^2-y^2.`

From product (i), we get

`P\\\\=(a+bi)(a-bi)\\\\=a^2-(bi)^2\\\\=a^2-b^2i^2\\\\=a^2-b^2* (-1)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }i^2=-1]\\\\=a^2+b^2.`

So, there is no imaginary part in the given product.

Thus, the correct option is

(D) The imaginary part is zero.