Show that the lines `(x-a+d)/(alpha-delta)=(y-a)/alpha=(z-a-d)/(alpha+delta)`and `(x-b+c)/(beta-gamma)=(y-b)/beta=(z-b-c)/(beta+gamma)`are coplanar.

Show that the lines (x-a+d)/(alpha-delta)=(y-a)/(alpha)=(z-a-d)/(alpha+delta)(x-b+c)/(beta-gamma)=(y-b)/(beta)=(z-b-c)/(beta+gamma) are coplanar.

Prove that the straight lines x/alpha=y/beta=z/gamma,x/l=y/m=z/n and x/(a alpha)=y/(b beta)=z/(c gamma) will be co planar if l/alpha(b-c)+m/beta(c-a)+n/gamma(a-b)=0

If |((beta+gamma-alpha -delta)^4 , (beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4, (gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4, (alpha + beta-gamma-delta)^2,1)|=-k(alpha -beta)(alpha -gamma)(alpha-delta)(beta-gamma)(beta-delta)(gamma-delta) , then the value of (k)^(1//2) is ____

If alpha,beta,gamma are different from 1 and are the roots of ax^(3)+bx^(2)+cx+d=0 and (beta-gamma)(gamma-alpha)(alpha-beta)=(25)/(2) ,then prove that det[[(alpha)/(1-alpha),(beta)/(1-beta),(gamma)/(1-gamma)(alpha)/(1-alpha),(beta)/(1-beta),(gamma)/(1-gamma)alpha,beta,gammaalpha^(2),beta^(2),gamma^(2)]]=(25d)/(2(a+b+c+d))

If alpha, beta, gamma and delta are the roots of the equation x ^(4) -bx -3 =0, then an equation whose roots are (alpha +beta+gamma)/(delta^(2)), (alpha +beta+delta)/(gamma^(2)), (alpha +delta+gamma)/(beta^(2)), and (delta +beta+gamma)/(alpha^(2)), is:

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|=-64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)