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If b is the harmonic mean of a and c and...

If b is the harmonic mean of a and c and `alpha ,beta` are the roots of the equation `a(b-c)x^2+b(c-a)x+c(a-b)=0`

Text Solution

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`b=(2ac)/(a+c)`
`alpha+beta=(-b(c-a))/(a(b-c))`
`=(ab-bc)/(ab-ac)`
`=(ab-bc)/(ab-(ab+bc)/2)`
`=(ab-bc)/((ab-bc)/2)`
`alpha+beta=2`
`alpha*beta=(c(a-b))/(a(b-c))`
`alpha*beta=(ac-bc)/(ab-ac)`
...
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