曲线和方程是平面解析几何中最基本的概念。曲线是具有某种性质的点的集合。曲线的方程就是曲线上的点所具有的共同性质在数量关系上的反映。曲线和方程是同一点集的两种不同的表现形式,曲线给出的是这点集的几何形象,而方程则给予解析式以说明,因此,只有当曲线与方程表示是同一点集时,才能说明曲线是方程的曲线,方程是该曲线的方程。在由给出曲线的条件推导曲线的方程时,往往由于不注意所给条件的各种可能性的研究,或者疏忽了限制条件的约束(许多时侯,这种约束条件是隐含的),因而导致缩小或者扩大了点集的范围,也由于要对方程进行化简整理,因而就可能破坏方程的同解性,使得在最后所得到的方程中,增加了一些不符合条件的部分,或者遗漏了合乎条件的部分,因而使得所得到的方程有一 Curves and equations are the most basic concepts in plane analytic geometry. A curve is a collection of points that have some kind of nature. The equation of the curve is the reflection of the quantitative nature of the common nature of the points on the curve. Curves and equations are two different representations of the same set of points. The curve gives the geometric image of the set, and the equation gives analytical expressions. Therefore, only when the curves and equations represent the same set of points, It can be explained that the curve is the curve of the equation and the equation is the equation of the curve. When deriving the equation of a curve from the conditions of a given curve, it is often due to the lack of attention to the study of the various possibilities of the given condition, or to the fact that the constraints of the constraints were neglected (many times, such constraints are implicit), As a result, the scope of the point set is reduced or expanded, and because the equations need to be sorted out, it is possible to break the equation’s symmetry and to add some unqualified parts to the resulting equation, or Missing the conditional part, thus resulting in the resulting equation