Question

If `alpha, beta, gamma` are the roots of the cubic `x^(3)-px^(2)+qx-r=0`
Find the equations whose roots are
(i) `beta gamma +1/(alpha), gamma alpha+1/(beta), alpha beta+1/(gamma)`
(ii)`(beta+gamma-alpha),(gamma+alpha-beta),(alpha+beta-gamma)`
Also find the valueof `(beta+gamma-alpha)(gamma+alpha-beta)(alpha+beta-gamma)`

Answer 1

Correct Answer - (i) `ry^(3)-q(r+1)y^(2)+p(r+1)^(2)y-(r+1)^(3)=0`
(ii) `y^(3)-py^(2)+(4q-p^(2))y+(8r-4pq+p^(3))=0` and `4pq-p^(3)-8r`
Given `alpha, beta` and `gamma` are the roots of the cubic equation
`x^(3)-px^(2)+qx-r=0` ………..i
`:. alpha +beta+gamma=p,alpha beta+beta gamma+gamma alpha=q,alpha beta gamma =r`
(i) Let `y=beta gamma +1/(alpha)`
`impliesy=(alpha beta gamma +1)/(alpha)=(r+1)/(alpha)`
`:.alpha=(r+1)/y`
From Eq. (i) we get
`alpha^(3)-palpha^(2)+q alpha-r=0`
`implies((r+1)^(3))/(y^(3))-(p(r+1)^(2))/(y^(2))+(q(r+1))/y-r=0`
or `ry^(3)-q(r+1)y^(2)+p(r+1)^(2)y-(r+1)^(3)=0`
(ii) Let `y=beta+gamma -alpha=(alpha+beta+gamma)-2alpha=p-2alpha`
`alpha=(p-y)/2`
From Eq. (i) we get
`alpha^(3)-palpha^(2)+q alpha-r=0`
`implies((p-y)^(3))/8-(p(p-y)^(2))/4+(q(p-y))/2-r=0`
or `y^(3)-py^(2)+(4q-p^(2))y+(8r-4pq+p^(3))=0`
Also product of roots `=(8r-4pq+p^(3))`