Задачі підвищеної складності: 41 (1)

41. Розв’яжіть систему рівнянь:

1) $\frac{3}{x^2+y^2-1}$ – $\frac{2y}{x}$ = 1, x² + y² = a, $\frac{3}{a-1}$ – $\frac{2}{b}$ = 1,

x² + y² + $\frac{4x}{y}$ = 22; $\frac{x}{y}$ = b; a + 4b = 22;

3b + 2a – 2 = ab – b, 4b + 2a – 2 – ab = 0,

a + 4b = 22 ; a ≠ 1, b ≠ 0; a = 22 – 4b;

4b + 44 – 8b – 2 – 22b + 4b² = 0, 4b² – 26b + 42 = 0,

a = 22 – 4b; a = 22 – 4b;

2b² – 13b + 21 = 0, b = 3, b = 3,5,

a = 22 – 4b; a = 22 – 4b.

Отже, маємо дві системи:

x² + y² = 10, x² + y² = 8,

$\frac{x}{y}$ = 3; $\frac{x}{y}$ = 3,5.

а) x² + y² = 10, 9x² + y² = 10, 10y² = 10, y² = 1,

$\frac{x}{y}$ = 3; x = 3y;  x = 3y; x = 3y;

x = 3, або х = –3,

y = 1 y = –1.

б) x² + y² = 8, 12,25y² + y² = 8, 13,25y² = 8, y = ±$\sqrt{\frac{8}{\mathrm{13,25}}}$,

$\frac{x}{y}$ = 3,5; x = 3,5y; x = 3,5y; x = 3,5y;

y = ±$\sqrt{\frac{8}{13\frac{1}{4}}}$, y = ±$\sqrt{8:\frac{53}{4}}$, y = ±$\sqrt{\frac{8•4}{53}}$, y = ±4$\sqrt{\frac{2}{53}}$,

x = 3,5y; x = 3,5y; x = 3,5y; x = 3,5y;

y = ±4$\sqrt{\frac{106}{53•53}}$, y = ±4 • $\frac{\sqrt{106}}{53}$, x = $\frac{7}{2}$ • $\frac{4\sqrt{106}}{53}$ = $\frac{14\sqrt{106}}{53}$,

x = 3,5y; x = 3,5y; y = $\frac{4\sqrt{106}}{53}$; або

x = –$\frac{14\sqrt{106}}{53}$,

y = –$\frac{4\sqrt{106}}{53}$.

Відповідь: (3; 1), (1; 3), ($\frac{14\sqrt{106}}{53}$; $\frac{14\sqrt{106}}{53}$), (–$\frac{14\sqrt{106}}{53}$; –$\frac{14\sqrt{106}}{53}$).

2) $\frac{3}{x^2+y^2}$ + 2xy = $\frac{21}{5}$, x² + y² = a, $\frac{1}{a}$ + b = $\frac{21}{5}$, b = $\frac{21}{5}$ – $\frac{1}{a}$,

$\frac{1}{2xy}$ + x² + y² = $\frac{21}{4}$; 2xy = b ; $\frac{1}{b}$ + a = $\frac{21}{4}$, $\frac{1}{b}$ + a = $\frac{21}{4}$,

b = $\frac{21a-5}{5a}$, b = $\frac{21a-5}{5a}$,

$\frac{5a}{21a-5}$ + a = $\frac{21}{4}$, 20a + a(21a – 5) • 4 = 21(21a – 5), a ≠ $\frac{5}{21}$;

b = $\frac{21a-5}{5a}$, b = $\frac{21a-5}{5a}$, b = $\frac{21a-5}{5a}$,

84a² = 441a – 105; 84a² – 441a + 105 = 0; 4a² – 21a + 5 = 0;

b = $\frac{21a-5}{5a}$, a = 5, або a = $\frac{1}{4}$;

a = 5, a = $\frac{1}{4}$; b = 4, b = $\frac{1}{5}$.

а) x² + y² = $\frac{1}{4}$, x² + y² = $\frac{1}{4}$, x² + y² = $\frac{1}{4}$, x² + $\frac{1}{{100x}^2}$ = $\frac{1}{4}$,

2xy = $\frac{1}{5}$; xy = $\frac{1}{10}$; y = $\frac{1}{10x}$; y = $\frac{1}{10x}$;

100x⁴ + 1 – 25x² = 0; 100x⁴ – 25x² + 1 = 0; x² = $\frac{1}{5}$, x² = $\frac{1}{20}$,

y = $\frac{1}{10x}$; y = $\frac{1}{10x}$; y = $\frac{1}{10x}$;

Тоді маємо чотири розв’язки:

x = $\frac{1}{\sqrt{5}}$, x = –$\frac{1}{\sqrt{5}}$, x = $\frac{1}{2\sqrt{5}}$, x = –$\frac{1}{2\sqrt{5}}$,

y = $\frac{\sqrt{5}}{10}$, y = –$\frac{\sqrt{5}}{10}$, y = $\frac{\sqrt{5}}{5}$, y = –$\frac{\sqrt{5}}{5}$.

б) x² + y² = 5, x² + y² = 5, x² + $\frac{4}{x^2}$ – 5 = 0, x⁴ – 5x² – 5 = 0;

2xy = 4; xy = 2; y = $\frac{2}{x}$; y = $\frac{2}{x}$;

Тоді маємо чотири розв’язки:

x = 2, x = –2, x = 1, x = –1;

y = 1; y = –1; y = 2; y = –2.

Відповідь: (2 ; 1), (–2 ; –1), (–1 ; –2), ($\frac{1}{\sqrt{5}}$ ; $\frac{\sqrt{5}}{10}$), (–$\frac{1}{\sqrt{5}}$ ; –$\frac{\sqrt{5}}{10}$), ($\frac{1}{2\sqrt{5}}$ ; $\frac{\sqrt{5}}{5}$), (–$\frac{1}{2\sqrt{5}}$ ; –$\frac{\sqrt{5}}{5}$).