設(shè)

A點坐標(biāo)(x1，y1)

B點坐標(biāo)(x2，y2)

有

x12/a2+y12/b2=1

x22/a2+y22/b2=1

x12+y12+x22+y22=a2+b2

即

x12+a2y12/b2=a2

b2x22/a2+y22=b2

即

x12+a2y12/b2+b2x22/a2+y22

=x12+y12+x22+y22

即

(a2-b2）y12/b2=(a2-b2）x22/a2

即

y1/x2=b/a

又

b2x12/a2+y12=b2

x22+a2y22/b2=a2

即

b2x12/a2+y12+x22+a2y22/b2

=x12+y12+x22+y22

即

(a2-b2）x12/a2=(a2-b2）y22/b2

即

y2/x1=b/a

即

y1y2/(x1x2)=b2/a2

即

x12+b2x22/a2+x22+b2x12/a2=a2+b2

即

(a2+b2)(x12+x22）/a2=a2+b2

即

x12+x22=a2

y12+y22=b2

即

(y12+y22）/(x12+x22)=b2/a2

即

(y2-y1)2/(x2-x1)2=b2/a2=y1y2/(x1x2)

即

kAB=±b/a=±√(kOA·kOB)

得證