Question

曲线f(x,y)=0关于直线x+y+C=0的对称曲线方程为 ,关于直线x-y+C=0的对称曲线方程为

Answer 1

曲线f(x,y)＝0关于直线x+y+C＝0的对称曲线方程为f(-C-y,-C-x)＝0,关于直线x-y+C=0的对称曲线方程为f(y-C,x+C)＝0.

设曲线f(x,y)＝0关于直线x+y+C＝0的对称曲线方程为g(x,y)＝0,点P(x,y)是g(x,y)＝0上任一点,点P(x,y)关于直线x+y+C＝0的对称点为Q(x0,y0),则根据轴对称的性质有(y0-y)/(x0-x)＝1,且(x+x0)/2+(y+y0)/2+C＝0,解得y0＝-C-x,x0＝-C-y.

又由已知得Q(x0,y0)在曲线f(x,y)＝0上,有f(x0,y0)＝0,即f(-C-y,-C-x)＝0.即g(x,y)＝f(-C-y,-C-x)＝0.

下同