Решите систему однородных уравнений Система: X^2-xy-20y^2=0 X^2+2y^2=27

1. Обозначим:

y = tx; {x^2 - xy - 20y^2 = 0;

{x^2 + 2y^2 = 27; {x^2 - tx^2 - 20t^2x^2 = 0;

{x^2 + 2t^2x^2 = 27; {x^2 (1 - t - 20t^2) = 0;

{x^2 (1 + 2t^2) = 27.

2. Решим уравнения:

x^2 (1 - t - 20t^2) = 0;

x = 0 не является решением системы.

1 - t - 20t^2 = 0; 20t^2 + t - 1 = 0; D = 1 ^2 + 4 * 20 = 81; t = (-1 ± √81) / 40 = (-1 ± 9) / 40;

1) t = (-1 - 9) / 40 = - 10/40 = - 1/4;

x^2 = 27 / (1 + 2t^2) = 27 / (1 + 2 * 1/16) = 27 / (1 + 1/8) = 27 * 8/9 = 24; x = ±√24 = ±2√6; y = tx = - 1/4 * (±2√6) = ∓√6/2.

2) t = (-1 + 9) / 40 = 8/40 = 1/5;

x^2 = 27 / (1 + 2t^2) = 27 / (1 + 2 * 1/25) = 27 / (27/25) = 25; x = ±√25 = ±5; y = tx = 1/5 * (±5) = ±1.

Ответ: (±2√6; ∓√6/2); (±5; ±1).