题目

Problem

Find the coordinates of the points of intersection of the curve and the line with equations

2xy + 5y^2 = 24 \quad \text{and} \quad 2x + y + 4 = 0.

[4]

题目中文翻译

求曲线与直线的交点坐标，它们的方程分别为

2xy + 5y^2 = 24 \quad \text{和} \quad 2x + y + 4 = 0

[4]

解答

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直线方程已经给出 $2x$ 与 $y$ 的关系。直接把 $2x=-y-4$ 代入曲线，求出 $y$ 后再回代。

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Substituting $2x=-y-4$ into $2xy+5y^2=24$ gives

(-y-4)y+5y^2=24 \quad\Longrightarrow\quad y^2-y-6=(y-3)(y+2)=0.

For $y=3$ , $x=-\frac72$ . For $y=-2$ , $x=-1$ . The intersections are $(-\frac72,3)$ and $(-1,-2)$ .