Question

若曲线y=x-1/2次方在点（a,a-1/2)处的切线与两个坐标轴围成的三角行的面积为18，则a=() A64 B32 C16 D8

Answer 1

y=x^(-1/2)
y'=-1/2x^(-3/2)
k=y'=-1/2a^(-3/2)
故切线方程是y-a^(-1/2)=-1/2a^(-3/2)*(x-a)=-1/2a^(-3/2)x+1/2a^(-1/2)
y=-1/2a^(-3/2)x+3/2a^(-1/2)
令X=0,Y=3/2a^(-1/2),令Y=0,X=3a
S=1/2*3a*3/2a^(-1/2)=18
a^1/2=8
a=64
选择A

Answer 2

y'=-1/2*x^(-3/2)
所以斜率-1/2*a^(-3/2)
则直线是y-a^(-1/2)=-1/2*a^(-3/2)(x-a)

y=0,
1=1/2*a^(-1)(x-a)
x-a=2a
x=3a

x=0
y-a^(-1/2)=-1/2*a^(-3/2)(-a)
y-a^(-1/2)=1/2*a^(-1/2)
y=3/2*a^(-1/2)

所以面积=|3a|*3/2*a^(-1/2)÷2=18
显然a>0
所以(9/4)a^(1/2)=18
a^(1/2)=8
a=64