Question Number 201581 by cortano12 last updated on 09/Dec/23
Answered by mr W last updated on 09/Dec/23
Commented by mr W last updated on 09/Dec/23
$${R}=\frac{\mathrm{8}+\mathrm{2}}{\mathrm{2}}=\mathrm{5} \\ $$$${r}={OM}=\sqrt{\mathrm{8}×\mathrm{2}}=\mathrm{4} \\ $$$${let}\:{x}=\frac{{PQ}}{\mathrm{2}} \\ $$$$\sqrt{{R}^{\mathrm{2}} −{x}^{\mathrm{2}} }+\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }={R} \\ $$$${r}^{\mathrm{2}} =\mathrm{2}{R}\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} } \\ $$$$\Rightarrow{x}={r}\sqrt{\mathrm{1}−\frac{{r}^{\mathrm{2}} }{\mathrm{4}{R}^{\mathrm{2}} }} \\ $$$$\Rightarrow{PQ}=\mathrm{2}{r}\sqrt{\mathrm{1}−\frac{{r}^{\mathrm{2}} }{\mathrm{4}{R}^{\mathrm{2}} }} \\ $$$$\:\:\:\:=\mathrm{2}×\mathrm{4}\sqrt{\mathrm{1}−\frac{\mathrm{4}^{\mathrm{2}} }{\mathrm{4}×\mathrm{5}^{\mathrm{2}} }}=\frac{\mathrm{8}\sqrt{\mathrm{21}}}{\mathrm{5}}\approx\mathrm{7}.\mathrm{332} \\ $$