(α,β);(β,ϒ) and (ϒ,α) are respectively the roots of x2 -2px+2 =0, x2 -2qx +3=0 and x2 -2rx+6=0.If α,β and ϒ are all positive ,then prove that  the value of p+q+r is 6

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Please find below the solution to the asked query:

Given α,β are the roots of x2-2px+2=0. So,α+β=2p     ....1αβ=2           ....2And β,γ are the roots of x2-2qx+3=0. So,β+γ=2q     ....3βγ=3           ....4And γ,α are the roots of x2-2rx+6=0. So,γ+α=2r     ....5γα=6           ....6Multiply equation 246, we get,αβγ2=2×3×6αβγ=6          ....7Now divide equation246 by equation7, we get,γ=3, α=2 and β=1Adding equation135, we get,α+β+β+γ+γ+α=2p+2q+2r2α+β+γ=2p+q+r22+1+3=2p+q+rp+q+r=6Hence Proved.

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