InterviewSolution
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InterviewSolution
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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)` |
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Answer» 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))` |
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