Question-172824

Question Number 172824 by Mikenice last updated on 01/Jul/22

Commented by MJS_new last updated on 02/Jul/22

how many % can solve this ?

I can .

Commented by cortano1 last updated on 02/Jul/22

o n e s o l u t i o n (0, 0)

Answered by mr W last updated on 02/Jul/22

l e t x = \sqrt{X} > 0, y = \sqrt{Y} > 0

2 (x^{2} - y^{2}) y = x \dots (i)

(x^{2} + y^{2}) x = 3 y \dots (i i)

x = y = 0 i s s o l u t i o n \Rightarrow X = Y = 0

(i) \times (i i) :

2 (x^{2} - y^{2}) (x^{2} + y^{2}) = 3

\Rightarrow x^{4} - y^{4} = \frac{3}{2} \dots (I)

(i) / (i i) :

\frac{2 (x^{2} - y^{2})}{x^{2} + y^{2}} = \frac{x^{2}}{3 y^{2}}

x^{4} - 5 x^{2} y^{2} + 6 y^{2} = 0

(x^{2} - 2 y^{2}) (x^{2} - 3 y^{2}) = 0 \dots (I I)

c a s e 1 : x^{2} = 2 y^{2}

\Rightarrow 4 y^{4} - y^{4} = \frac{3}{2}

\Rightarrow y^{4} = \frac{1}{2} \Rightarrow y^{2} = \frac{1}{\sqrt{2}} \Rightarrow Y = \frac{1}{\sqrt{2}}

\Rightarrow x^{2} = \sqrt{2} \Rightarrow X = \sqrt{2}

c a s e 2 : x^{2} = 3 y^{2}

\Rightarrow 9 y^{4} - y^{4} = \frac{3}{2}

\Rightarrow y^{4} = \frac{3}{16} \Rightarrow y^{2} = \frac{\sqrt{3}}{4} \Rightarrow Y = \frac{\sqrt{3}}{4}

\Rightarrow x^{2} = \frac{3 \sqrt{3}}{4} \Rightarrow X = \frac{3 \sqrt{3}}{4}

s u m m a r y :

(X, Y) = (0, 0), (\sqrt{2}, \frac{1}{\sqrt{2}}), (\frac{3 \sqrt{3}}{4}, \frac{\sqrt{3}}{4})

Commented by Tawa11 last updated on 02/Jul/22

Great sir

Answered by MJS_new last updated on 02/Jul/22

{\begin{cases} 2 y^{1 / 2} x - x^{1 / 2} - 2 y^{3 / 3} = 0 \\ x^{3 / 2} + {y x}^{1 / 2} - 3 y^{1 / 2} = 0 \end{cases}

y = p x

{\begin{cases} \sqrt{x} (2 \sqrt{p} (p - 1) x + 1) = 0 \\ \sqrt{x} ((p + 1) x - 3 \sqrt{p}) = 0 \end{cases}

\Rightarrow x_{1} = 0 \Rightarrow y_{1} = 0

{\begin{cases} x = - \frac{1}{2 \sqrt{p} (p - 1)} \\ x = \frac{3 \sqrt{p}}{p + 1} \end{cases}

\Rightarrow

p^{2} - \frac{5}{6} p + \frac{1}{6} = 0

\Rightarrow

p_{2} = \frac{1}{3} \Rightarrow x_{2} = \frac{3 \sqrt{3}}{4} \Rightarrow y_{2} = \frac{\sqrt{3}}{4}

p_{3} = \frac{1}{2} \Rightarrow x_{3} = \sqrt{2} \Rightarrow y_{3} = \frac{\sqrt{2}}{2}

Commented by Tawa11 last updated on 03/Jul/22

Great sir

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