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Question-49740




Question Number 49740 by ajfour last updated on 09/Dec/18
Commented by ajfour last updated on 09/Dec/18
Find side length s of a pentagon  of equal sides inscribed within  the ellipse, in terms of ellipse  parameters a and b.
$${Find}\:{side}\:{length}\:\boldsymbol{{s}}\:{of}\:{a}\:{pentagon} \\ $$$${of}\:{equal}\:{sides}\:{inscribed}\:{within} \\ $$$${the}\:{ellipse},\:{in}\:{terms}\:{of}\:{ellipse} \\ $$$${parameters}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}}. \\ $$
Answered by ajfour last updated on 10/Dec/18
let   A(h,k), B(0,b) ,  E((s/2), −b(√(1−(s^2 /(4a^2 )))) )  AB = AE = s  ⇒  h^2 +(b−k)^2  = s^2     &         (h^2 /a^2 )+(k^2 /b^2 ) = 1      (h−(s/2))^2 +(k+b(√(1−(s^2 /(4a^2 )))))^2 = s^2   .....
$${let}\:\:\:{A}\left({h},{k}\right),\:{B}\left(\mathrm{0},{b}\right)\:, \\ $$$${E}\left(\frac{{s}}{\mathrm{2}},\:−{b}\sqrt{\mathrm{1}−\frac{{s}^{\mathrm{2}} }{\mathrm{4}{a}^{\mathrm{2}} }}\:\right) \\ $$$${AB}\:=\:{AE}\:=\:{s} \\ $$$$\Rightarrow\:\:{h}^{\mathrm{2}} +\left({b}−{k}\right)^{\mathrm{2}} \:=\:{s}^{\mathrm{2}} \:\:\:\:\& \\ $$$$\:\:\:\:\:\:\:\frac{{h}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{k}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=\:\mathrm{1} \\ $$$$\:\:\:\:\left({h}−\frac{{s}}{\mathrm{2}}\right)^{\mathrm{2}} +\left({k}+{b}\sqrt{\mathrm{1}−\frac{{s}^{\mathrm{2}} }{\mathrm{4}{a}^{\mathrm{2}} }}\right)^{\mathrm{2}} =\:{s}^{\mathrm{2}} \\ $$$$….. \\ $$
Commented by ajfour last updated on 10/Dec/18
Any way out, please..
$${Any}\:{way}\:{out},\:{please}.. \\ $$

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