Question Number 80731 by ajfour last updated on 05/Feb/20
Commented by ajfour last updated on 05/Feb/20
$${The}\:{pentagon}\:{has}\:{each}\:{side}\:{s}. \\ $$$${Find}\:{s},\:{in}\:{terms}\:{of}\:{ellipse} \\ $$$${parameters}\:{a}\:{and}\:{b}. \\ $$
Answered by mr W last updated on 06/Feb/20
$${A}\left({a}\:\mathrm{cos}\:\alpha,\:{b}\:\mathrm{sin}\:\alpha\right) \\ $$$${B}\left({a}\:\mathrm{cos}\:\beta,\:−{b}\:\mathrm{sin}\:\beta\right) \\ $$$$\left({a}\:\mathrm{cos}\:\alpha\right)^{\mathrm{2}} +\left({b}−{b}\:\mathrm{sin}\:\alpha\right)^{\mathrm{2}} ={s}^{\mathrm{2}} \\ $$$$\left({a}\:\mathrm{cos}\:\alpha−{a}\:\mathrm{cos}\:\beta\right)^{\mathrm{2}} +\left({b}\:\mathrm{sin}\:\alpha+{b}\:\mathrm{sin}\:\beta\right)^{\mathrm{2}} ={s}^{\mathrm{2}} \\ $$$${a}\:\mathrm{cos}\:\beta=\frac{{s}}{\mathrm{2}} \\ $$$${let}\:\frac{{b}}{{a}}=\mu,\:\frac{{s}}{{a}}=\lambda \\ $$$$\mathrm{cos}^{\mathrm{2}} \:\alpha+\mu^{\mathrm{2}} \left(\mathrm{1}−\mathrm{sin}\:\alpha\right)^{\mathrm{2}} =\lambda^{\mathrm{2}} \:\:\:…\left({i}\right) \\ $$$$\left(\mathrm{cos}\:\alpha−\mathrm{cos}\:\beta\right)^{\mathrm{2}} +\mu^{\mathrm{2}} \left(\mathrm{sin}\:\alpha+\mathrm{sin}\:\beta\right)^{\mathrm{2}} =\lambda^{\mathrm{2}} \:\:\:…\left({ii}\right) \\ $$$$\mathrm{cos}\:\beta=\frac{\lambda}{\mathrm{2}}\:\:\:…\left({iii}\right) \\ $$$$ \\ $$$$\left(\mathrm{1}−\mu^{\mathrm{2}} \right)\mathrm{sin}^{\mathrm{2}} \:\alpha+\mathrm{2}\mu^{\mathrm{2}} \:\mathrm{sin}\:\alpha−\left(\mathrm{1}+\mu^{\mathrm{2}} −\lambda^{\mathrm{2}} \right)=\mathrm{0} \\ $$$${if}\:\mu=\mathrm{1}: \\ $$$$\mathrm{sin}\:\alpha=\mathrm{1}−\frac{\lambda^{\mathrm{2}} }{\mathrm{2}} \\ $$$${if}\:\mu<\mathrm{1}: \\ $$$$\Rightarrow\mathrm{sin}\:\alpha=\frac{\mu^{\mathrm{2}} −\sqrt{\mathrm{1}−\left(\mathrm{1}−\mu^{\mathrm{2}} \right)\lambda^{\mathrm{2}} }}{\mathrm{1}−\mu^{\mathrm{2}} } \\ $$$$\Rightarrow\alpha=\mathrm{sin}^{−\mathrm{1}} \left[\frac{\mu^{\mathrm{2}} −\sqrt{\mathrm{1}−\left(\mathrm{1}−\mu^{\mathrm{2}} \right)\lambda^{\mathrm{2}} }}{\mathrm{1}−\mu^{\mathrm{2}} }\right] \\ $$$$\Rightarrow\beta=\mathrm{cos}^{−\mathrm{1}} \frac{\lambda}{\mathrm{2}} \\ $$$${put}\:{them}\:{into}\:\left({i}\right)\:{to}\:{get}\:\lambda. \\ $$$$ \\ $$$${examples}: \\ $$$$\mu=\frac{{b}}{{a}}=\mathrm{1}:\:\lambda=\frac{{s}}{{a}}=\mathrm{1}.\mathrm{1756} \\ $$$$\mu=\frac{{b}}{{a}}=\frac{\mathrm{2}}{\mathrm{3}}:\:\lambda=\frac{{s}}{{a}}=\mathrm{1}.\mathrm{3312} \\ $$
Commented by ajfour last updated on 06/Feb/20
$${let}\:{me}\:{some}\:{time}\:{to}\:{follow},\:{Sir}. \\ $$