Question Number 196637 by sonukgindia last updated on 28/Aug/23
Answered by AST last updated on 28/Aug/23
$${a}=\sqrt{\mathrm{12}−\sqrt{{m}}};{b}=\sqrt{\mathrm{12}+\sqrt{{m}}},{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{24} \\ $$$${ab}=\sqrt{\mathrm{144}−{m}};\left({a}+{b}\right)^{\mathrm{2}} =\mathrm{24}+\mathrm{2}\sqrt{\mathrm{144}−{m}} \\ $$$$\Rightarrow{a}+{b}=\sqrt{\mathrm{24}+\mathrm{2}\sqrt{\mathrm{144}−{m}}}={n} \\ $$$$\Rightarrow{p}^{\mathrm{2}} =\mathrm{144}−{m}\geqslant\mathrm{0}\Rightarrow{n}=\sqrt{\mathrm{24}+\mathrm{2}{p}}\Rightarrow\mathrm{24}+\mathrm{2}{p}={x}^{\mathrm{2}} \\ $$$${p}\in\left\{\mathrm{1},\mathrm{2},..,{n},..\mathrm{11}\right\}\Rightarrow{p}=\mathrm{6}\Rightarrow{n}=\mathrm{6}\Rightarrow{m}=\mathrm{108} \\ $$$$\Rightarrow{m}+{n}=\mathrm{114} \\ $$