设曲线的参数方程为x=a(cost+tsint)y=a(sint-tcost)求证原点到曲线上所有点的法线距离相等

Question

设曲线的参数方程为 x=a(cost+tsint) y=a(sint-tcost) 求证原点到曲线上所有点的法线距离相等

Answer 1

x=a(cost+tsint),y=a(sint-tcost),
∴x'=a(-sint+sint+tcost)=atcost,
y'=a(cost-cost+tsint)=atsint,
∴法线l斜率=-x'/y'=-cost/sint,
法线l方程：y-a(sint-tcost)=(-cost/sint)[x-a(cost+tsint)],
即xcost+ysint-a=0,
∴原点到l的距离=|a|,与t无关,命题成立.