4xy-1-4-1-x-2-y-1-y-2-1-Resolver-elsistema-en-R-

Question Number 72488 by aliesam last updated on 29/Oct/19

{\begin{cases} 4 x y = 1 \\ 4 \sqrt{1 - x^{2}} (y - \sqrt{1 - y^{2}}) = 1 \end{cases}

R e s o l v e r e l s i s t e m a e n R

Answered by behi83417@gmail.com last updated on 30/Oct/19

x = cost, y = cosr

\Rightarrow {\begin{cases} 4 sint (cosr - sinr) = 1 \\ 4 cost . cosr = 1 \end{cases}

\Rightarrow \frac{1}{{(cosr - sinr)}^{2}} + \frac{1}{\cos^{2} r} = 4 \Rightarrow

\Rightarrow \cos^{2} r + {(cosr - sinr)}^{2} = {4 \cos}^{2} r . {(cosr - sinr)}^{2}

\frac{1 + \cos 2 r}{2} + 1 - \sin 2 r = 2 (1 + \cos 2 r) (1 - \sin 2 r)

\Rightarrow 1 + \cos 2 r + 2 - 2 \sin 2 r =

4 (1 - \sin 2 r + \cos 2 r - \sin 2 rcos 2 r)

\Rightarrow 3 + \cos 2 r - 2 \sin 2 r =

4 - 4 \sin 2 r + 4 \cos 2 r - 4 \sin 2 rcos 2 r

\Rightarrow 3 \cos 2 r - 2 \sin 2 r - 4 \sin 2 rcos 2 r + 1 = 0

[let : c = \cos 2 r, s = \sin 2 r] \Rightarrow

3 c - 2 s - 4 sc + 1 = 0

{9 c}^{2} + {4 s}^{2} - 12 sc = {16 s}^{2} c^{2} - 8 sc + 1

{9 c}^{2} + {4 s}^{2} - {16 s}^{2} c^{2} = 4 sc + 1

{5 c}^{2} - {16 c}^{2} (1 - c^{2}) + 3 = 4 sc

\Rightarrow {16 c}^{4} - {11 c}^{2} + 3 = 4 sc

\Rightarrow {256 c}^{8} + {121 c}^{4} + 9 - {352 c}^{6} + {96 c}^{4} - {66 c}^{2} =

= {16 c}^{2} - {16 c}^{4}

\Rightarrow {256 c}^{8} - {352 c}^{6} + {233 c}^{4} - {82 c}^{2} + 9 = 0

\Rightarrow c = \pm 0.42, \pm 0.76

\Rightarrow \cos 2 r = \pm 0.42, \pm 0.76

\Rightarrow {\begin{cases} \cos 2 r = 0.42 \Rightarrow y = cosr = 0.843 \\ \Rightarrow x = \frac{1}{4 y} = \frac{1}{4 \times 0.843} = 0.211 \end{cases}

\Rightarrow {\begin{cases} \cos 2 r = - 0.42 \Rightarrow y = cosr = 0.54 \\ x = \frac{1}{4 y} = \frac{1}{4 \times 0.211} = 0.053 \end{cases}

\Rightarrow {\begin{cases} \cos 2 r = 0.76 \Rightarrow y = cosr = 0.938 \\ x = \frac{1}{4 y} = \frac{1}{4 \times 0.938} = 0.235 \end{cases}

\Rightarrow {\begin{cases} \cos 2 r = - 0.76 \Rightarrow y = cosr = 0.346 \\ x = \frac{1}{4 y} = \frac{1}{4 \times 0.346} = 0.087 \end{cases}

Commented by aliesam last updated on 29/Oct/19

g o d b l e s s y o u

Answered by MJS last updated on 29/Oct/19

(1) \Rightarrow x = \frac{1}{4 y}

\Rightarrow

(2) \frac{\sqrt{16 y^{2} - 1} (y - \sqrt{1 - y^{2}})}{∣ y ∣} = 1

\sqrt{16 y^{2} - 1} (y - \sqrt{1 - y^{2}}) - ∣ y ∣= 0

\Rightarrow - 1 ⩽ y ⩽ - \frac{1}{4} \lor \frac{1}{4} ⩽ y ⩽ 1

we can only approximate (it leads to a

biquartic in y \Rightarrow quartic in \sqrt{y} which has

no “ {nice}^{″} solution)

the only solution I get is

x \approx . 302667; y \approx . 825989

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