Страница 257 номер 986, ГДЗ по алгебре за 9 класс к учебнику Мерзляка

1) Прямой \(3x - y - 5 = 0\) и параболы \(y = \frac{2}{3}x^{2} - 2x + 4\)
\(\begin{cases} 3x - y - 5 = 0 \\ y = \frac{2}{3}x^{2} - 2x + 4 \end{cases}\ \Rightarrow \begin{cases} y = 3x - 5 \\ 3x - 5 = \frac{2}{3}x^{2} - 2x + 4 \end{cases}\ \Rightarrow\)
\(3x - 5 = \dfrac{2}{3}x^{2} - 2x + 4\)
\(\dfrac{2}{3}x^{2} - 5x + 9 = 0\)
\(2x^{2} - 15x + 27 = 0\)
\(D = 225 - 4 \cdot 2 \cdot 27 = 225 - 216 = 9\)
\(x_{1} = \dfrac{15 - 3}{4} = \dfrac{12}{4} = 3;\ y_{1} = 3 \cdot 3 - 5 = 4\)
\(x_{2} = \dfrac{15 + 3}{4} = \dfrac{18}{4} = \dfrac{9}{2};\ y_{2} = 3 \cdot \dfrac{9}{2} - 5 = \dfrac{27 - 10}{2} = \dfrac{17}{2} = 8{,}5\)

Точки пересечения \((3;\ 4);\ (4{,}5;\ 8{,}5)\)

2) Прямой \(2x - 3y - 3 = 0\) и гиперболы \(xy = 3\)
\(\left\{ \begin{aligned} & 2x - 3y - 3 = 0 \\ & xy = 3 \end{aligned} \right.\ \Rightarrow \left\{ \begin{aligned} & x = \dfrac{3y + 3}{2} \\ & y \cdot \dfrac{3}{2} \cdot (y + 1) = 3 \end{aligned} \right.\ \Rightarrow \begin{cases} x = \frac{3y + 3}{2} \\ y^{2} + y = 2 \end{cases}\)
\(y^{2} + y - 2 = 0\)
\((y - 1)(y + 2) = 0\)
\(y_{1} = 1;\ x_{1} = \dfrac{3 \cdot 1 + 3}{2} = 3\)
\(y_{2} = - 2;\ x_{2} = \dfrac{3 \cdot ( - 2) + 3}{2} = - 1{,}5\)

Точки пересечения \((3;\ 1);\ ( - 1{,}5; - 2)\)

3) Окружности \(x^{2} + y^{2} = 13\) и гиперболы \(xy = 6\).
\(\left\{ \begin{aligned} & x^{2} + y^{2} = 13 \\ & xy = 6 \end{aligned} \right.\ \Rightarrow \left\{ \begin{aligned} & x^{2} + \dfrac{36}{x^{2}} = 13 \\ & y = \dfrac{6}{x} \end{aligned} \right.\ \Rightarrow \left| t = x^{2} \right| \Rightarrow\)
\(\Rightarrow \begin{cases} t + \frac{36}{t} = 13 \\ y = \frac{6}{x} \end{cases}\)
\(t^{2} + 36 - 13t = 0\)
\((t - 4)(t - 9) = 0\)
\(t = 4 \Rightarrow x^{2} = 4 \Rightarrow \begin{cases} x_{1} = 2;\ y_{1} = 3 \\ x_{2} = - 2;\ y_{2} = - 3 \end{cases}\)
\(t = 9 \Rightarrow x^{2} = 9 \Rightarrow \begin{cases} x_{3} = 3;\ y_{3} = 2 \\ x_{4} = - 3;\ y_{4} = - 2 \end{cases}\)

Точки пересечения \((2;\ 3);\ ( - 2; - 3);\ (3;\ 2);\ ( - 3; - 2)\)

Ответ: 1) \((3;\ 4);\ (4{,}5;\ 8{,}5)\); 2) \((3;\ 1);\ ( - 1{,}5; - 2)\); 3) \((2;\ 3);\ ( - 2; - 3);\ (3;\ 2);\ ( - 3; - 2)\).