拋物線x^2=(1/4)y上點M到y(tǒng)=4x-5的距離最短,則過拋物線x^2=(1/4)y點M的切線,與直線y=4x-5平行,設(shè)過M的直線為:
y=4x+b
代入x^2=(1/4)y,得
x^2=(1/4)y=(1/4)*(4x+b)
x^2-x-(5/4)b=0
過點M的直線y=4x+b與拋物線x^2=(1/4)y相切,則上方程的判別式△=0,即
△=(-1)^2-4*1*[-(5/4)b]=0
b=-1/5
過點M的直線y=4x-1/5
現(xiàn)在問題變?yōu)榍髢芍本€y=4x-5與y=4x-1/5的距離:
y=4x-5
x=0,y=-5,N(0,-5)
點N(0,-5)到y(tǒng)=4x-1/5的距離
L=|4*0-(-5)-1/5|/√17 =4.8/√17
M到y(tǒng)=4x-5的最短距離=4.8/√17