Question Number 189684 by normans last updated on 20/Mar/23
Commented by a.lgnaoui last updated on 21/Mar/23
Answered by mr W last updated on 20/Mar/23
Commented by mr W last updated on 21/Mar/23
$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\left(\sqrt{\mathrm{5}}{a}\right)^{\mathrm{2}} =\mathrm{5}{a}^{\mathrm{2}} \\ $$$$\left(\mathrm{16}−{y}−{y}\right)^{\mathrm{2}} +\mathrm{22}^{\mathrm{2}} =\left(\mathrm{2}\sqrt{\mathrm{5}}{a}\right)^{\mathrm{2}} \\ $$$${y}^{\mathrm{2}} −\mathrm{16}{y}+\mathrm{185}−\mathrm{5}{a}^{\mathrm{2}} =\mathrm{0}\:\:\:…\left({i}\right) \\ $$$$\Rightarrow{y}=\mathrm{8}−\sqrt{\mathrm{5}{a}^{\mathrm{2}} −\mathrm{121}} \\ $$$$\left(\mathrm{22}−{x}−{x}\right)^{\mathrm{2}} +\mathrm{16}^{\mathrm{2}} =\left(\mathrm{4}{a}\right)^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} −\mathrm{22}{x}+\mathrm{185}−\mathrm{4}{a}^{\mathrm{2}} =\mathrm{0}\:\:\:…\left({ii}\right) \\ $$$$\Rightarrow{x}=\mathrm{11}−\sqrt{\mathrm{4}{a}^{\mathrm{2}} −\mathrm{64}} \\ $$$$\left({i}\right)+\left({ii}\right): \\ $$$$\mathrm{11}{x}+\mathrm{8}{y}=\mathrm{185}−\mathrm{2}{a}^{\mathrm{2}} \\ $$$$\mathrm{11}\left(\mathrm{11}−\sqrt{\mathrm{4}{a}^{\mathrm{2}} −\mathrm{64}}\right)+\mathrm{8}\left(\mathrm{8}−\sqrt{\mathrm{5}{a}^{\mathrm{2}} −\mathrm{121}}\right)=\mathrm{185}−\mathrm{2}{a}^{\mathrm{2}} \\ $$$$\mathrm{11}\sqrt{\mathrm{4}{a}^{\mathrm{2}} −\mathrm{64}}=\mathrm{2}{a}^{\mathrm{2}} −\mathrm{8}\sqrt{\mathrm{5}{a}^{\mathrm{2}} −\mathrm{121}} \\ $$$${a}^{\mathrm{2}} −\mathrm{41}=\mathrm{8}\sqrt{\mathrm{5}{a}^{\mathrm{2}} −\mathrm{121}} \\ $$$${a}^{\mathrm{4}} −\mathrm{402}{a}^{\mathrm{2}} +\mathrm{10705}=\mathrm{0} \\ $$$$\Rightarrow{a}^{\mathrm{2}} =\mathrm{201}−\mathrm{32}\sqrt{\mathrm{29}}\approx\mathrm{28}.\mathrm{675} \\ $$