Find the acute angle between the lines `(x-1)/l=(y+1)/m=1/na n d=(x+1)/m=(y-3)/n=(z-1)/lw h e r el > m > n ,a n dl ,m ,n` are the roots of the cubic equation `x^3+x^2-4x=4.`

Explanation

Given lines are `(x-1)/(l)=(y+1)/(m)=(z)/(n) and (x+1)/(m)=(y-3)/(n)=(z-1)/(l)`
Angle between lines is gives by
`" "costheta=(lm+mn+nl)/(l^(2)+m^(2)+n^(2))" "`(i)
Now `l, m, n` are the roots of the cubic equation `x^(3)+x^(2)-4x=4`. Thus,
`" "l+m+n=-1 and lm+mn+nl=-4`
`" "(l+m+n)^(2)=l^(2)+m^(2)+n^(2)+2(lm+mn+nl)`
or `" "(-1)^(2)=l^(2)+m^(2)+n^(2)+2(-4)`
or `" "l^(2)+m^(2)+n^(2)=9`
`therefore" "costheta=-(4)/(9)`
Therefore, acute angle between the lines is `cos^(-1)""(4)/(9)`.