Question Number 21137 by Tinkutara last updated on 14/Sep/17
$$\mathrm{If}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mid{z}+\mathrm{1}+{i}\mid\:+\:\mid{z}−\mathrm{1}−{i}\mid\:+\:\mid\mathrm{2}\:−\:{z}\mid\:+\:\mid\mathrm{3}\:−\:{z}\mid\:\mathrm{is} \\ $$$${k}\:\mathrm{then}\:\left({k}\:−\:\mathrm{8}\right)\:\mathrm{equals} \\ $$
Answered by Tinkutara last updated on 15/Sep/17
$$\mid{z}_{\mathrm{1}} \mid+\mid{z}_{\mathrm{2}} \mid\geqslant\mid{z}_{\mathrm{1}} +{z}_{\mathrm{2}} \mid \\ $$$${So}\:{minimum}\:{value}\:{is} \\ $$$$\mid{z}+\mathrm{1}+{i}+{z}−\mathrm{1}−{i}+\mathrm{2}−{z}+\mathrm{3}−{z}\mid=\mathrm{5}={k} \\ $$$${k}−\mathrm{8}=−\mathrm{3} \\ $$