Question Number 53828 by ajfour last updated on 26/Jan/19 | ||
Commented by ajfour last updated on 26/Jan/19 | ||
$${Find}\:{a}\:{and}\:{b}\:{of}\:{maximum}\:{area}\:{ellipse} \\ $$$${within}\:{the}\:{trapezium}. \\ $$ | ||
Answered by ajfour last updated on 26/Jan/19 | ||
Commented by ajfour last updated on 26/Jan/19 | ||
$${Let}\:{B}\left({h},{k}\right)\:,\:\:\mathrm{tan}\:\alpha=\mathrm{2} \\ $$$${C}\left({h}+\mathrm{5}{s}\mathrm{cos}\:\theta,{k}+\mathrm{5}{s}\mathrm{sin}\:\theta\right) \\ $$$${A}\left({h}−\mathrm{4}{s}\mathrm{sin}\:\theta,{k}+\mathrm{4}{s}\mathrm{cos}\:\theta\right) \\ $$$${D}\left({h}−\mathrm{4}{s}\mathrm{sin}\:\theta+\mathrm{3}{s}\mathrm{cos}\:\theta,\:{k}+\mathrm{4}{s}\mathrm{cos}\:\theta+\mathrm{3}{s}\mathrm{sin}\:\theta\right) \\ $$$${let}\:\mathrm{tan}\:\theta={m} \\ $$$${y}_{{int},{BC}} ={k}−{mh} \\ $$$${y}_{{int},{AB}} ={k}+\frac{{h}}{{m}} \\ $$$${y}_{{int},{AD}} ={k}+\mathrm{4}{s}\mathrm{cos}\:\theta−{m}\left({h}−\mathrm{4}{s}\mathrm{sin}\:\theta\right) \\ $$$${y}_{{int},{CD}} ={k}+\mathrm{5}{s}\mathrm{sin}\:\theta−\left(\frac{{m}−\mathrm{2}}{\mathrm{1}+\mathrm{2}{m}}\right)\left({h}+\mathrm{5}{s}\mathrm{cos}\:\theta\right) \\ $$$${due}\:{to}\:{tangency}\:{of}\:{AB},{BC},{CD},{AD} \\ $$$$\:\:{k}−{mh}=−\sqrt{{a}^{\mathrm{2}} {m}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:\:\:\:\:\:\:\:....\left({i}\right) \\ $$$$\:\:{k}+\frac{{m}}{{h}}\:=\:−\sqrt{\frac{{a}^{\mathrm{2}} }{{m}^{\mathrm{2}} }+{b}^{\mathrm{2}} }\:\:\:\:\:\:\:\:....\left({ii}\right) \\ $$$${k}+\mathrm{4}{s}\mathrm{cos}\:\theta−{m}\left({h}−\mathrm{4}{s}\mathrm{sin}\:\theta\right)=\sqrt{{a}^{\mathrm{2}} {m}^{\mathrm{2}} +{b}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\left({iii}\right) \\ $$$${k}+\mathrm{5}{s}\mathrm{sin}\:\theta−\left(\frac{{m}−\mathrm{2}}{\mathrm{1}+\mathrm{2}{m}}\right)\left({h}+\mathrm{5}{s}\mathrm{cos}\:\theta\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\sqrt{{a}^{\mathrm{2}} \left(\frac{{m}−\mathrm{2}}{\mathrm{1}+\mathrm{2}{m}}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} }\:\:\:\:\:...\left({iv}\right) \\ $$$${And}\:\:{for}\:{maximum}\:{area}\:{of}\:{ellipse} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{d}\left({ab}\right)=\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\left({v}\right) \\ $$$${Unknowns}\:{are}\:\:\:{h},{k},{m},{a},{b}\:. \\ $$ | ||