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Question
If the ratio of the roots of the equation
lx2 + nx + n = 0 is p: q, Prove that
`sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.`
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Solution
Let α, β be the roots of
lx2 + nx + n = 0, α + β = `n/l` and αβ = `n/l`.
`α/beta = p/q ...("given")`
Now L.H.S.
= `sqrt(p/q) + sqrt(q/p) + sqrt(n/l)`
= `sqrt(α/beta) + sqrt(beta/α) + sqrt(n/l)`
= `(α + beta)/sqrt(αbeta) + sqrt(n/l)`
= `((-n)/l)/sqrt(n/l) + sqrt(n/l,) [ ∵ α + beta = (-n)/l αbeta = n/l]`
= `-sqrt(n/l) + sqrt(n/l)`
= 0
= R.H.S.
Hence proved.
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