△ABC的三个顶点在椭圆4X^2+5 y^2=6上,其中AB两点关于原点O对称,设直线AC的斜率k1,直线BC的斜率k2,则k1k2的值为
第1个回答 2020-09-11
易知a=√6/2,b=√30/5
设A(√6/2cosα,√30/5sinα),则B(-√6/2cosα,-√30/5sinα)
设C(√6/2cosβ,√30/5sinβ)
则k1=[√30/5(sinβ-sinα)]/[√6/2(cosβ-cosα)]=2√5/5*(sinβ-sinα)/(cosβ-cosα)
k2=[√30/5(sinβ+sinα)]/[√6/2(cosβ+cosα)]=2√5/5*(sinβ+sinα)/(cosβ+cosα)
∴k1k2=4/5*(sin²β-sin²α)/(cos²β-cos²α)=4/5*(cos2α-cos2β)/(cos2β-cos2α)=-4/5
设A(√6/2cosα,√30/5sinα),则B(-√6/2cosα,-√30/5sinα)
设C(√6/2cosβ,√30/5sinβ)
则k1=[√30/5(sinβ-sinα)]/[√6/2(cosβ-cosα)]=2√5/5*(sinβ-sinα)/(cosβ-cosα)
k2=[√30/5(sinβ+sinα)]/[√6/2(cosβ+cosα)]=2√5/5*(sinβ+sinα)/(cosβ+cosα)
∴k1k2=4/5*(sin²β-sin²α)/(cos²β-cos²α)=4/5*(cos2α-cos2β)/(cos2β-cos2α)=-4/5
