Question

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

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

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

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

Thus, the correct option is

(D) The imaginary part is zero.