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Question Number 49737 by ajfour last updated on 09/Dec/18

Commented by ajfour last updated on 09/Dec/18

Find parameters a and b of  ellipse circumscribing a rectangle  of sides l and h.

$${Find}\:{parameters}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}}\:{of} \\ $$$${ellipse}\:{circumscribing}\:{a}\:{rectangle} \\ $$$${of}\:{sides}\:\boldsymbol{{l}}\:{and}\:\boldsymbol{{h}}. \\ $$

Answered by ajfour last updated on 09/Dec/18

(l^2 /a^2 )+(h^2 /b^2 ) = 4  ,  let T = a^2 b^2   T = a^2 ((h^2 /(4−(l^2 /a^2 )))) = ((a^4 h^2 )/(4a^2 −l^2 ))  (dT/da) = ((4h^2 a^3 (4a^2 −l^2 )−8a^5 h^2 )/((4a^2 −l^2 )^2 ))  (dT/da) = 0  ⇒   a = (l/(√2))   ,  b = (h/(√2)) .

$$\frac{{l}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{h}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=\:\mathrm{4}\:\:,\:\:{let}\:{T}\:=\:{a}^{\mathrm{2}} {b}^{\mathrm{2}} \\ $$$${T}\:=\:{a}^{\mathrm{2}} \left(\frac{{h}^{\mathrm{2}} }{\mathrm{4}−\frac{{l}^{\mathrm{2}} }{{a}^{\mathrm{2}} }}\right)\:=\:\frac{{a}^{\mathrm{4}} {h}^{\mathrm{2}} }{\mathrm{4}{a}^{\mathrm{2}} −{l}^{\mathrm{2}} } \\ $$$$\frac{{dT}}{{da}}\:=\:\frac{\mathrm{4}{h}^{\mathrm{2}} {a}^{\mathrm{3}} \left(\mathrm{4}{a}^{\mathrm{2}} −{l}^{\mathrm{2}} \right)−\mathrm{8}{a}^{\mathrm{5}} {h}^{\mathrm{2}} }{\left(\mathrm{4}{a}^{\mathrm{2}} −{l}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$\frac{{dT}}{{da}}\:=\:\mathrm{0}\:\:\Rightarrow\:\:\:{a}\:=\:\frac{{l}}{\sqrt{\mathrm{2}}}\:\:\:,\:\:{b}\:=\:\frac{{h}}{\sqrt{\mathrm{2}}}\:. \\ $$

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