Question Number 131196 by john_santu last updated on 02/Feb/21
$$\:{The}\:{minimum}\:{value}\:{of}\:{the}\: \\ $$$${expression}\:{B}\:=\:\mid{z}\mid^{\mathrm{2}} +\mid{z}−\mathrm{3}\mid^{\mathrm{2}} +\mid{z}−\mathrm{6}{i}\mid^{\mathrm{2}} \\ $$$${is}\:{p}.\:{What}\:{the}\:{value}\:{of}\:\frac{{p}}{\mathrm{10}}\:.? \\ $$
Answered by liberty last updated on 02/Feb/21
$${z}={a}+{bi}\:\rightarrow\begin{cases}{\mid{z}\mid^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\\{\mid{z}−\mathrm{3}\mid^{\mathrm{2}} =\:\left({a}−\mathrm{3}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} \:}\\{\mid{z}−\mathrm{6}{i}\mid^{\mathrm{2}} \:=\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{36}−\mathrm{12}{b}}\end{cases} \\ $$$${now}\:{we}\:{find}\: \\ $$$${B}=\mathrm{3}{a}^{\mathrm{2}} +\mathrm{3}{b}^{\mathrm{2}} −\mathrm{6}{a}−\mathrm{12}{b}+\mathrm{45}\: \\ $$$${B}=\mathrm{3}\left({a}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{3}\left({b}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{30} \\ $$$${so}\:{the}\:{minimum}\:{value}\:{of}\:{B}\:\mathrm{is}\:\mathrm{30}\:\mathrm{when} \\ $$$$\:\begin{cases}{{a}=\mathrm{1}}\\{{b}=\mathrm{2}}\end{cases}\:\Rightarrow\:{z}=\mathrm{1}+\mathrm{2}{i}\:{then}\:{p}\:=\:\mathrm{30}\: \\ $$$${and}\:\frac{{p}}{\mathrm{10}}\:=\:\frac{\mathrm{30}}{\mathrm{10}}\:=\:\mathrm{3}. \\ $$$$ \\ $$
Commented by john_santu last updated on 02/Feb/21
$${gave}\:{kudos}\: \\ $$