Question

|COMPLEX NUMBERS| Determine the complex conjugate of... ? Thx!

`((a+bi)/(a -bi))^2 - ((a-bi)/(a +bi))^2`

Answer 1

See below.

Explanation:

If `z = x+i y` then `bar z = x-iy` so

`bar(((a+bi)/(a -bi))^2 - ((a-bi)/(a +bi))^2) = ((a-bi)/(a +bi))^2 - ((a+bi)/(a -bi))^2 = (8ab(b^2- a^2 ))/(a^2+b^2)^2i`

Answer 2

Complex conjugate is `-(8ab(a^2-b^2)i)/((a^2+b^2)^2`

Explanation:

`((a+bi)/(a-bi))^2-((a-bi)/(a+bi))^2`

= `(a^2-b^2+2abi)/(a^2-b^2-2abi)-(a^2-b^2-2abi)/(a^2-b^2+2abi)`

= `((a^2-b^2)^2+4ab(a^2-b^2)i-4a^2b^2-((a^2-b^2)^2-4ab(a^2-b^2)i-4a^2b^2))/((a^2-b^2)^2+4a^2b^2)`

= `(8ab(a^2-b^2)i)/((a^2+b^2)^2`

Hence complex conjugate is `-(8ab(a^2-b^2)i)/((a^2+b^2)^2)`