Question Number 53492 by ajfour last updated on 22/Jan/19
Commented by ajfour last updated on 22/Jan/19
$${Find}\:{parameters}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}}\:{of}\:{an} \\ $$$${ellipse}\:{that}\:{circumscribes}\:{an} \\ $$$${equilateral}\:{triangle}\:{of}\:{side}\:\boldsymbol{{t}}\:{and} \\ $$$${even}\:{a}\:{square}\:{of}\:{side}\:\boldsymbol{{s}}. \\ $$
Answered by mr W last updated on 22/Jan/19
$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$$\frac{{s}^{\mathrm{2}} }{\mathrm{4}{a}^{\mathrm{2}} }+\frac{{s}^{\mathrm{2}} }{\mathrm{4}{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$$\Rightarrow\frac{\mathrm{1}}{{a}^{\mathrm{2}} }+\frac{\mathrm{1}}{{b}^{\mathrm{2}} }=\frac{\mathrm{4}}{{s}^{\mathrm{2}} } \\ $$$$\frac{{t}^{\mathrm{2}} }{\mathrm{4}{a}^{\mathrm{2}} }+\frac{\left(\frac{\sqrt{\mathrm{3}}{t}}{\mathrm{2}}−{b}\right)^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$$\Rightarrow\frac{{t}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{\left(\sqrt{\mathrm{3}}{t}−\mathrm{2}{b}\right)^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{4} \\ $$$$\Rightarrow\frac{\mathrm{4}{t}^{\mathrm{2}} }{{s}^{\mathrm{2}} }−\frac{{t}^{\mathrm{2}} }{{b}^{\mathrm{2}} }+\left(\sqrt{\mathrm{3}}\frac{{t}}{{b}}−\mathrm{2}\right)^{\mathrm{2}} =\mathrm{4} \\ $$$${with}\:\mu=\frac{{t}}{{s}},\:\lambda=\frac{{t}}{{b}} \\ $$$$\Rightarrow\mathrm{4}\mu^{\mathrm{2}} −\lambda^{\mathrm{2}} +\left(\sqrt{\mathrm{3}}\lambda−\mathrm{2}\right)^{\mathrm{2}} =\mathrm{4} \\ $$$$\Rightarrow\lambda^{\mathrm{2}} −\mathrm{2}\sqrt{\mathrm{3}}\lambda+\mathrm{2}\mu^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\lambda=\frac{{t}}{{b}}=\sqrt{\mathrm{3}}+\sqrt{\mathrm{3}−\mathrm{2}\mu^{\mathrm{2}} } \\ $$$$\Rightarrow\frac{{s}}{{b}}×\mu=\sqrt{\mathrm{3}}+\sqrt{\mathrm{3}−\mathrm{2}\mu^{\mathrm{2}} } \\ $$$$\Rightarrow{b}=\frac{{t}}{\:\sqrt{\mathrm{3}}+\sqrt{\mathrm{3}−\mathrm{2}\mu^{\mathrm{2}} }} \\ $$$$\Rightarrow{a}=\frac{{s}}{\:\sqrt{\mathrm{4}−\left(\frac{{s}}{{b}}\right)^{\mathrm{2}} }}=\frac{{s}}{\:\sqrt{\mathrm{4}−\left(\frac{\sqrt{\mathrm{3}}+\sqrt{\mathrm{3}−\mathrm{2}\mu^{\mathrm{2}} }}{\mu}\right)^{\mathrm{2}} }} \\ $$
Commented by ajfour last updated on 23/Jan/19
$${Its}\:{shorter}\:{than}\:{the}\:{parametric}\:{way}, \\ $$$${thanks}\:{Sir}. \\ $$