Question Number 107415 by ajfour last updated on 10/Aug/20
Commented by ajfour last updated on 10/Aug/20
$${Find}\:\:{r}/{R}. \\ $$
Answered by mr W last updated on 10/Aug/20
Commented by mr W last updated on 10/Aug/20
$${BA}=\mathrm{2}{R}−{r} \\ $$$${OD}={R}−\mathrm{2}{r} \\ $$$$\frac{{OD}}{{BC}}=\frac{{OA}}{{BA}} \\ $$$$\Rightarrow\frac{{R}−\mathrm{2}{r}}{{r}}=\frac{{R}}{\mathrm{2}{R}−{r}} \\ $$$$\left(\frac{{R}}{{r}}−\mathrm{2}\right)\left(\mathrm{2}−\frac{{r}}{{R}}\right)=\mathrm{1} \\ $$$${let}\:\lambda=\frac{{R}}{{r}} \\ $$$$\left(\lambda−\mathrm{2}\right)\left(\mathrm{2}\lambda−\mathrm{1}\right)=\lambda \\ $$$$\Rightarrow\lambda^{\mathrm{2}} −\mathrm{3}\lambda+\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow\lambda=\frac{\mathrm{3}+\sqrt{\mathrm{5}}}{\mathrm{2}}\approx\mathrm{2}.\mathrm{618} \\ $$
Commented by ajfour last updated on 10/Aug/20
$${Very}\:{very}\:{nice}\:{solution}\:{Sir},\:{thanks}! \\ $$