Question Number 159392 by ajfour last updated on 16/Nov/21
Commented by ajfour last updated on 16/Nov/21
$$\:\:{Find}\:{radius}\:{of}\:{the}\:{circles}\left({equal}\right). \\ $$
Answered by mr W last updated on 17/Nov/21
Commented by mr W last updated on 18/Nov/21
$${eqn}.\:{of}\:{OC}: \\ $$$${say}\:{y}={mx}\:\Rightarrow{mx}−{y}=\mathrm{0} \\ $$$${eqn}.\:{of}\:{AD}: \\ $$$${sau}\:{y}=−{k}\left({x}−\mathrm{1}\right)\:\Rightarrow{kx}+{y}−{k}=\mathrm{0} \\ $$$${P}\left({r},\mathrm{1}−{r}\right) \\ $$$${Q}\left({q},{r}\right) \\ $$$$ \\ $$$${r}=−\frac{{mr}−\mathrm{1}+{r}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }}\:\:\:…\left({i}\right) \\ $$$${r}=\frac{{mq}−{r}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }}\:\:\:…\left({ii}\right) \\ $$$${r}=−\frac{{kr}+\mathrm{1}−{r}−{k}}{\:\sqrt{\mathrm{1}+{k}^{\mathrm{2}} }}\:\:\:…\left({iii}\right) \\ $$$${r}=−\frac{{kq}+{r}−{k}}{\:\sqrt{\mathrm{1}+{k}^{\mathrm{2}} }}\:\:\:…\left({iv}\right) \\ $$$$\mathrm{4}\:{unknowns}:\:{r},\:{m},\:{k},\:{q} \\ $$$$ \\ $$$${mq}−{r}+{mr}−\mathrm{1}+{r}=\mathrm{0} \\ $$$$\Rightarrow{q}=\frac{\mathrm{1}}{{m}}−{r} \\ $$$${kq}+{r}−{k}={kr}+\mathrm{1}−{r}−{k} \\ $$$$\Rightarrow{k}=\frac{\mathrm{1}−\mathrm{2}{r}}{{q}−{r}} \\ $$$${k}=\frac{\mathrm{1}−\mathrm{2}{r}}{\frac{\mathrm{1}}{{m}}−\mathrm{2}{r}}\:\:\:…\:\left({I}\right) \\ $$$$ \\ $$$${from}\:\left({ii}\right): \\ $$$${r}=\frac{\mathrm{1}−\left({m}+\mathrm{1}\right){r}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }} \\ $$$${r}=\frac{\mathrm{1}}{\mathrm{1}+{m}+\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }}\:\:\:…\left({II}\right) \\ $$$${from}\:\left({iv}\right): \\ $$$$\mathrm{1}−{k}+\sqrt{\mathrm{1}+{k}^{\mathrm{2}} }=\left(\mathrm{1}−\frac{\mathrm{1}}{{m}}\right)\frac{{k}}{{r}}\:\:\:…\left({III}\right) \\ $$$${putting}\:{I},\:{II}\:{into}\:{III}\:{we}\:{get}\:{an}.\:{eqn}. \\ $$$${for}\:{m}. \\ $$$$\Rightarrow{m}\approx\mathrm{1}.\mathrm{243756} \\ $$$$\Rightarrow{r}\approx\mathrm{0}.\mathrm{2694} \\ $$
Commented by mr W last updated on 18/Nov/21
Commented by ajfour last updated on 18/Nov/21
$${Excellent}\:{sir},\:{thanks}! \\ $$$${i}\:{m}\:{out}\:{of}\:{m}\:{town}\:{to}\:{take}\:{up}\:{a}\:{job}, \\ $$$${hence}\:{my}\:{inconsistency}.. \\ $$
Commented by mr W last updated on 18/Nov/21
$${good}\:{luck}\:{with}\:{the}\:{new}\:{job}\:{sir}! \\ $$
Commented by Tawa11 last updated on 28/Nov/21
$$\mathrm{Great}\:\mathrm{sir} \\ $$