a-x-a-y-2a-a-x-a-y-2a-a-R-

Question Number 62228 by behi83417@gmail.com last updated on 17/Jun/19

{\begin{cases} \sqrt{a + x} + \sqrt{a - y} = 2 a \\ \sqrt{a - x} + \sqrt{a + y} = 2 a \end{cases} a \in R .

Commented by mr W last updated on 18/Jun/19

a > 0

- a ⩽ x, y ⩽ a

x = y

\sqrt{a + x} + \sqrt{a - x} = 2 a

\sqrt{a + x} = 2 a - \sqrt{a - x}

a + x = 4 a^{2} - 4 a \sqrt{a - x} + a - x

x = 2 a^{2} - 2 a \sqrt{a - x}

2 a^{2} - x = 2 a \sqrt{a - x}

4 a^{4} - 4 a^{2} x + x^{2} = 4 a^{3} - 4 a^{2} x

x^{2} = 4 (1 - a) a^{3}

\Rightarrow x = y = \pm 2 a \sqrt{a (1 - a)}

\Rightarrow a ⩽ 1

2 a \sqrt{a (1 - a)} ⩽ a

\Rightarrow a ⩾ \frac{1}{2}

i . e . s o l u t i o n e x i s t s o n l y w h e n

\frac{1}{2} ⩽ a ⩽ 1 .

Commented by behi83417@gmail.com last updated on 18/Jun/19

thank you very much dear master .