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If the roots of the equation : (b-c)x^...

If the roots of the equation `:`
`(b-c)x^(2) + (c-a) x + ( a-b) = 0 ` are equal, then a,b,c are in `:`

Text Solution

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Give equation is
`a(b-c)x^(2)+b(c-a)x+c(a-b)=0`…………i
Here, coefficient of `x^(2)+` coefficient of `x+` constant term `=0`
i.e. `a(b-c)+b(c-a)+c(a-b)=0`
The, 1 is a root of Eq. (i)
Since, its roots are equal.
Therefore its other root will be also equal to 1.
Then product of roots `=1xx1=(c(a-b))/(a(b-c))`
`impliesab-ac=ca-bc`
`:.b=(2ac)/(a+c)`
Hence a,b,c are in HP.
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