2 Complex numbers questions: If z is complex and |z+16|=4|z+1|, find |z|. Solve -i/(x-iy) = (4+7i)/(5-3i)

[umricky]

I attached a pic of the questions below, thanks in advance!

 

[khansaheb]

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[pka]

View attachment 36739

Suppose that [imath]s~\&~t[/imath] are real numbers and that [imath]z=s+ti[/imath].
Then for #4, [imath]z+16=(s+16)+ti[/imath] so that [imath]|z+16|=\sqrt{(s+16)^2+t^2}~\&~4|z+1|=4\sqrt{(s+1)^2+t^2}[/imath]

#6 [imath]\dfrac{-i}{x-yi}=\dfrac{4+7i}{5-3i}[/imath] is equivalent to [imath](-i)(5-3i)=(x-yi)(4+7i)[/imath].
Recall that the real parts are equal and that the imagery parts are equal.
[imath][/imath][imath][/imath][imath][/imath]