Question

x^2+根号3 y=根号5,y^2+根号3 x=根号5,且x≠y,求y/x+x/y

Answer 1

由已知：x^2+√3y=√5,y^2+√3x=√5,所以：x^2+√3y=y^2+√3x

即：x^2-y^2+√3y-√3x=0,所以：(x+y)(x-y)-√3(x-y)=0

因为：x≠y,所以：x+y=√3,

则：y=√3-x,x=√3-y,

分别代入x^2+√3y=√5,y^2+√3x=√5中,得到：x^2-√3x+3-√5=0,y^2-√3y+3-√5=0

即：x,y分别为方程A^2-√3A+3-√5=0的两个根,

所以：x+y=√3,xy=3-√5,

所以：y/x+x/y=(x^2+y^2)/(xy)=[(x+y)^2-2xy]/(xy)=(x+y)^2/(xy)-2=3/(3-√5)-2

=3/4*(3+√5)-2=(3√5+1)/4.