Question Number 101159 by hardylanes last updated on 30/Jun/20
$${given}\:{the}\:{complex}\:{number}\:{z}\:{such}\:{that} \\ $$$${z}−\mathrm{4}{i}={a}+\mathrm{3}{zi}.\: \\ $$$${find}\:{the}\:{value}\:{of}\:{a}\:{if}\:\:{z}\:{is}\:{purwly}\:{imaginary} \\ $$$$ \\ $$
Answered by Rio Michael last updated on 01/Jul/20
$${z}−\mathrm{3}{zi}\:=\:{a}\:+\:\mathrm{4}{i} \\ $$$$\Rightarrow\:{z}\:=\:\frac{{a}\:+\:\mathrm{4}{i}}{\mathrm{1}−\mathrm{3}{i}} \\ $$$$\Rightarrow\:{z}\:=\:\frac{\left({a}\:+\:\mathrm{4}{i}\right)\left(\mathrm{1}\:+\:\mathrm{3}{i}\right)}{\left(\mathrm{1}\:+\:\mathrm{3}{i}\right)\left(\mathrm{1}−\mathrm{3}{i}\right)}\:=\:\frac{{a}\:+\:\mathrm{3}{ai}\:+\:\mathrm{4}{i}−\mathrm{12}}{\mathrm{10}} \\ $$$$\Rightarrow\:\:{z}\:=\:\frac{{a}−\mathrm{12}}{\mathrm{10}}\:+\:\frac{\mathrm{3}{a}\:+\:\mathrm{4}}{\mathrm{10}}{i} \\ $$$$\mathrm{If}\:{z}\:\mathrm{is}\:\mathrm{purely}\:\mathrm{imaginary},\:\mathrm{then}\:{Re}\left({z}\right)\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\:\frac{{a}−\mathrm{12}}{\mathrm{10}}\:=\:\mathrm{0}\:\:\Leftrightarrow\:\:{a}\:=\:\mathrm{12} \\ $$
Commented by hardylanes last updated on 01/Jul/20
thanks