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

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

2 Answers
Jan 10, 2018

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#

Jan 11, 2018

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)#