Question Number 228497 by ajfour last updated on 18/Apr/26
Answered by TonyCWX last updated on 18/Apr/26
![OD^2 = x^4 −c^2 (1/(x^4 −c^2 )) + (1/x^2 ) = 1 x^2 + x^4 − c^2 = x^6 − c^2 x^2 c^2 (x^2 −1) = x^6 −x^4 −x^2 c^2 = ((x^6 −x^4 −x^2 )/(x^2 −1)) (1/x) = (x^2 /(c+x)) ⇒ c+x = x^3 ⇒ c = x^3 −x (x^3 −x)^2 = ((x^6 −x^4 −x^2 )/(x^2 −1)) x^8 −3x^6 +3x^4 −x^2 = x^6 −x^4 −x^2 x^8 −4x^6 +4x^4 = 0 x^4 (x^4 −4x^2 +4) = 0 x^4 (x^2 −2)^2 = 0 x = 0[REJECTED] or x = ±(√2) x > 0 ⇒ x = (√2) c^2 = ((((√2))^6 −((√2))^4 −((√2))^2 )/(((√2))^2 −1)) = ((8−4−2)/(2−1)) = 2 ⇒ c = (√2)](https://www.tinkutara.com/question/Q228498.png)
$${OD}^{\mathrm{2}} \:=\:{x}^{\mathrm{4}} −{c}^{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{{x}^{\mathrm{4}} −{c}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:=\:\mathrm{1} \\ $$$${x}^{\mathrm{2}} \:+\:{x}^{\mathrm{4}} \:−\:{c}^{\mathrm{2}} \:=\:{x}^{\mathrm{6}} \:−\:{c}^{\mathrm{2}} {x}^{\mathrm{2}} \\ $$$${c}^{\mathrm{2}} \left({x}^{\mathrm{2}} −\mathrm{1}\right)\:=\:{x}^{\mathrm{6}} −{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \\ $$$${c}^{\mathrm{2}} \:=\:\frac{{x}^{\mathrm{6}} −{x}^{\mathrm{4}} −{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} −\mathrm{1}} \\ $$$$ \\ $$$$\frac{\mathrm{1}}{{x}}\:=\:\frac{{x}^{\mathrm{2}} }{{c}+{x}}\:\Rightarrow\:{c}+{x}\:=\:{x}^{\mathrm{3}} \:\Rightarrow\:{c}\:=\:{x}^{\mathrm{3}} −{x} \\ $$$$\left({x}^{\mathrm{3}} −{x}\right)^{\mathrm{2}} \:=\:\frac{{x}^{\mathrm{6}} −{x}^{\mathrm{4}} −{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} −\mathrm{1}} \\ $$$${x}^{\mathrm{8}} −\mathrm{3}{x}^{\mathrm{6}} +\mathrm{3}{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \:=\:{x}^{\mathrm{6}} −{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \\ $$$${x}^{\mathrm{8}} −\mathrm{4}{x}^{\mathrm{6}} +\mathrm{4}{x}^{\mathrm{4}} \:=\:\mathrm{0} \\ $$$${x}^{\mathrm{4}} \left({x}^{\mathrm{4}} −\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}\right)\:=\:\mathrm{0} \\ $$$${x}^{\mathrm{4}} \left({x}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$${x}\:=\:\mathrm{0}\left[\boldsymbol{\mathrm{REJECTED}}\right]\:\boldsymbol{\mathrm{or}}\:{x}\:=\:\pm\sqrt{\mathrm{2}} \\ $$$${x}\:>\:\mathrm{0}\:\Rightarrow\:{x}\:=\:\sqrt{\mathrm{2}} \\ $$$$ \\ $$$${c}^{\mathrm{2}} \:=\:\frac{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{6}} −\left(\sqrt{\mathrm{2}}\right)^{\mathrm{4}} −\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} }{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{1}}\:=\:\frac{\mathrm{8}−\mathrm{4}−\mathrm{2}}{\mathrm{2}−\mathrm{1}}\:=\:\mathrm{2} \\ $$$$\Rightarrow\:{c}\:=\:\sqrt{\mathrm{2}} \\ $$