a-b-b-a-b-3-b-a-a-a-b-2-a-b-R-

Question Number 65495 by behi83417@gmail.com last updated on 30/Jul/19

{\begin{cases} \frac{a}{b} + \frac{b}{a + b} = \sqrt{3} \\ \frac{b}{a} + \frac{a}{a + b} = \sqrt{2} \end{cases} [a, b \in R]

Answered by MJS last updated on 31/Jul/19

b = a t

{\begin{cases} \frac{t^{2} + t + 1}{t (t + 1)} = \sqrt{3} \\ \frac{t^{2} + t + 1}{t + 1} = \sqrt{2} \end{cases}

{\begin{cases} t^{2} + t - \frac{1 + \sqrt{3}}{2} = 0 \\ t^{2} + (1 - \sqrt{2}) t + 1 - \sqrt{2} = 0 \end{cases}

{\begin{cases} t = - \frac{1}{2} \pm \frac{\sqrt{3 + 2 \sqrt{3}}}{2} \\ t = - \frac{1 - \sqrt{2}}{2} \pm \frac{\sqrt{- 1 + 2 \sqrt{2}}}{2} \end{cases}

\Rightarrow no solution

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