抛物线标准方程推导~抛物线标准方程中,x^2=2py,x^2=-2py的推导

Question

抛物线标准方程推导~
抛物线标准方程中,x^2=2py,x^2=-2py的推导

Answer 1

推导x^2=2py：
设点M(x,y)到直线y=-p/2的距离,和到点F(0,p/2)的距离相等.
点M(x,y)到直线y=-p/2的距离=[y+p/2],[MF]=根号[x^2+(y-p/2)^2].
[y+p/2]^2=x^2+(y-p/2)^2 y^2+py+p^2/4=x^2+y^2-py+p^2/4 x^2=2py
推导x^2=-2py：
设点M(x,y)到直线y=p/2的距离,和到点F(0,-p/2)的距离相等.
点M(x,y)到直线y=p/2的距离=[y-p/2],[MF]=根号[x^2+(y+p/2)^2].
[y-p/2]^2=x^2+(y+p/2)^2 y^2-py+p^2/4=x^2+y^2+py+p^2/4 x^2=-2py