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`, then

A

`alpha + beta = 3`

B

`alpha+beta=(1)/(2)`

C

`alpha beta = 2`

D

`alpha = 1, beta = 1`

Text Solution

Verified by Experts

The correct Answer is:

D

We observe that the sum of the coefficients of the given equation is zero. Therefore, 1 is a root of the given equation.
Let `alpha = 1 and beta` be the other root. Then,
`beta xx 1 =(c(a-b))/(a(b-c))rArr beta = (c)/(a) xx {(a-(2ac)/(a+c))/((2ac)/(a+c)-c)}=1" "[because b = (2ac)/(a+c)]`
Hence, `alpha = beta = 1`

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