郭敦顒回答:
直线l:y=[(1/2)√3] x,过点A(0,1)作AB⊥Y轴交l于B,作BA1⊥l交Y轴于A1;作A1B1⊥Y轴交l于B1,作B1A2⊥l交Y轴于A2;
作A2B2⊥Y轴交l于B2,作B2A3⊥l交Y轴于A3;
作A3B3⊥Y轴交l于B3,作B3A4⊥l交Y轴于A4;
则A4的坐标是
y=1时,x=1/[(1/2)√3]= (2/3)√3,B的坐标是B((2/3)√3,1),
BA1的斜率k=-1//[(1/2)√3]= -(2/3)√3,BA1的直线方程按点斜式有:
y-1=-[(2/3)√3][ x-(2/3)√3]= -[(2/3)√3] x+4/3,
y=-[(2/3)√3] x+7/3,
x=0时,y=7/3,A1的坐标是A1(0,7/3);
在y=[(1/2)√3] x中,
当y=7/3时,x=(7/3)/[(1/2)√3]=(7/3)(2/3)√3=(14/9)√3,
B1的坐标是B1((14/9)√3,7/3);
B1A2的直线方程是:y-7/3=-[(2/3)√3][ x-(14/9)√3]
=-[(2/3)√3] x+28/9,
y=-[(2/3)√3] x+49/9,
x=0时,y=49/9,A2的坐标是A2(0,49/9);
在y=[(1/2)√3] x中,
当y=49/9时,x=(49/9)/[(1/2)√3]=(49/9)(2/3)√3=(98/27)√3,
B2的坐标是B2((98/27)√3,49/9);
B2A3的直线方程是:y-49/9=-[(2/3)√3][ x-(98/27)√3]
=-[(2/3)√3] x+(196/81)
y =-[(2/3)√3] x+637/81,
x=0时,y=637/81,A3的坐标是A3(0,637/81);
在y=[(1/2)√3] x中,
当y= 637/81时,x=(637/81)/[(1/2)√3]=(637/81)(2/3)√3
=(1274/243)√3,
B3的坐标是B3((1274/243)√3,637/81);
B3A4的直线方程是:y-637/81=-[(2/3)√3][ x-(1274/243)√3],
=-[(2/3)√3] x+2548/729,
y=-[(2/3)√3] x+8281/729,
x=0时,y=(8281/729)√3,
A4的坐标是A4(0,8281/729).