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Question-201581




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=((8+2)/2)=5  r=OM=(√(8×2))=4  let x=((PQ)/2)  (√(R^2 −x^2 ))+(√(r^2 −x^2 ))=R  r^2 =2R(√(r^2 −x^2 ))  ⇒x=r(√(1−(r^2 /(4R^2 ))))  ⇒PQ=2r(√(1−(r^2 /(4R^2 ))))      =2×4(√(1−(4^2 /(4×5^2 ))))=((8(√(21)))/5)≈7.332
$${R}=\frac{\mathrm{8}+\mathrm{2}}{\mathrm{2}}=\mathrm{5} \\ $$$${r}={OM}=\sqrt{\mathrm{8}×\mathrm{2}}=\mathrm{4} \\ $$$${let}\:{x}=\frac{{PQ}}{\mathrm{2}} \\ $$$$\sqrt{{R}^{\mathrm{2}} −{x}^{\mathrm{2}} }+\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }={R} \\ $$$${r}^{\mathrm{2}} =\mathrm{2}{R}\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} } \\ $$$$\Rightarrow{x}={r}\sqrt{\mathrm{1}−\frac{{r}^{\mathrm{2}} }{\mathrm{4}{R}^{\mathrm{2}} }} \\ $$$$\Rightarrow{PQ}=\mathrm{2}{r}\sqrt{\mathrm{1}−\frac{{r}^{\mathrm{2}} }{\mathrm{4}{R}^{\mathrm{2}} }} \\ $$$$\:\:\:\:=\mathrm{2}×\mathrm{4}\sqrt{\mathrm{1}−\frac{\mathrm{4}^{\mathrm{2}} }{\mathrm{4}×\mathrm{5}^{\mathrm{2}} }}=\frac{\mathrm{8}\sqrt{\mathrm{21}}}{\mathrm{5}}\approx\mathrm{7}.\mathrm{332} \\ $$

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