试证明x^(2/3)+y^(2/3)=a^(2/3)(a>0)的切线夹在坐标轴之间长度等于常数a

Question

试证明 x^(2/3)+y^(2/3)=a^(2/3)(a>0)的切线夹在坐标轴之间长度等于常数a

Answer 1

星形线的参数方程是x=a(cosθ)^3,y=a(sinθ)^3,
在点（a(cosθ)^3,a(sinθ)^3）处：
dy/dx=[3a(sinθ)^2cosθ]/[3a(cosθ)^2(-sinθ)]=-tanθ
切线方程为[y-a(sinθ)^3]/[x-a(cosθ)^3]=-tanθ,
与x轴交于点P,令切线方程中y=0,得P的横坐标为Xp=acosθ,
与y轴交于点Q,令切线方程中x=0,得Q的纵坐标为Yq=asinθ,
PQ^2=(asinθ)^2+(acosθ)^2=a^2=const.