Find two complex numbers such that their sum is $\ 1+4i$ , their quotient is purely imaginary, and the real part of one of them is$\ -1 $.
I have tried to create two complex values called: $\ z=-1+ci $ and $\ w=a+bi $ and computing its sum knowing the result. Therefore: $\ -1+ci+a+bi=1+4i \rightarrow a=2 \land c+b=4 $.
Then, we know that $\ \frac{-1+ci}{2+bi}=xi $ being$\ x $ the variable that I included to form every purely imaginary complex number. So, here is when i got stuck.
Thank you in advance!
As has being answered, you just need to use the relation $\ c=4-b \land z=xiw $. Solving for b will output a value of $\ b=2+\sqrt2 \lor b=2-\sqrt2 $ resulting in $\ c=2-\sqrt2 $ if $\ b=2+\sqrt2 $ and $\ c=2+\sqrt2 $ if $\ b=2-\sqrt2 $.
Its just a matter of patience (so don't be like me, be patient).
Because $b+c=4$, we may as well write $c=4-b$. Hence you have the purely imaginary $$\frac{-1+(4-b)i}{2+bi}=\frac{-1+(4-b)i}{2+bi}\frac{2-bi}{2-bi}=\cdots$$