Question

已知曲线C:f(x)=x+a/x(a>0),直线l:y=x,在曲线C上有一个动点P,过点P分别作直线l和y轴的垂线,垂足

Answer 1

不影响结果,不妨设P(p,p + a/p)在第一象限,p > 0

B(0,p + a/p)

y = x的斜率为1,PA的斜率为-1,PA的方程:y - (p + a/p) = -(x - p)

令y = x,x - (p + a/p) = p - x

x = y = p + a/(2p)

A(p + a/(2p),p + a/(2p))

三角形ABP的面积 = (1/2)BP*BP上的高

= (1/2)p[p + a/p - p - a/(2p)]

= a/4 = 1/2

a = 2

f'(x) = 1 - a/x²

f'(p) = 1 - a/p²

过点P曲线C的切线:y - (p + a/p) = (1 - a/p²)(x - p)

令x = 0,y = 2a/p,N(0,2a/p)

令y = x,y = x = 2p,M(2p,2p)

三角形OMN的面积 = (1/2)ON*M的横坐标

= (1/2)(2a/p)(2p)

= 2a

= 4