Find the conditions if roots of the equation `x^(3)-px^(2)+qx-r=0` are in
(i) AP (ii)GP
(iii) HP

Explanation

(i) Let roots of the given equation are
`A-D,A,A+D` then
`A-D+A+A+D=pimpliesA=p/3`
Now A is the roots of the given equation, then it must be satisfy
`A^(3)-pA^(2)+qA-r=0`
`implies(p/3)^(2)-p(p/3)^(2)+q(p/3)-r=0`
`impliesp^(3)-3p^(3)+9qp-27r=0`
or `2p^(3)-9pQ=27r=0`
which is the required condition.
(ii) Let roots fo the given equation are `A/R, A,AR` then
`A/R. A. AR=(-1)^(3).(-r/1)=r`
`impliesA^(3)=r`
`impliesA=r^(1/3)`
Now A is the roots of the givenn equation, then
`A^(3)-pA^(2)+qA-r=0`
`impliesr-p(r)^(2//3)=q(r)^(1//3)-r=0`
or `p(r)^(2//3)=q(r)^(1//3)`
or `p^(3)r^(2)=q^(3)r`
or `p^(3)r=q^(3)`
which is the required condition.
(iii) Given equation is
`x^(3)-px^(2)+qx-r=0`...........i
On replacing `x` by `1/x` in Eq. (i) then
`(1/x)^(3)-p(1/x)^(2)+q(1/x)-r=0`
`impliesrx^(3)-qx^(2)+px-1=0`..........ii
Now roots of Eq. (ii) are in AP.
Let roots of Eq. (ii) are `A-P,A,A+P`, then
`A-P+A+A+p=q/r` or `A=q/(3r)`
`:.A` is a root of Eq. (ii) then
`rA^(3)-qA^(2)+pA-1=0`
`impliesr(1/(34))^(3)-q(q/(3r))^(2)+p(q/(3r))-1=0`
`impliesq^(3)-3q^(3)+9pqr-27r^(2)=0`
`implies2q^(3)-9pqr+27r^(2)=0`
which is the required condition.