If a, b, c are in H.P., then the equation `a(b-c) x^(2) + b(c-a)x+c(a-b)=0`

Options

A. has real and distinct roots
B. has equal roots
C. has no real root
D. none of these

Answer

B

Explanation

We know, that `a(b-c) + a(b-c)+b(c-a)+c(a-b) = 0`.
So, x = 1 is a root of the given equation.
Let `alpha` be the other root. Then,
`alpha xx 1 = (c(a-b))/(a(b-c)) rArr alpha = ((1)/(b)-(1)/(a))/((1)/(c)-(1)/(b))`
`rArr" "alpha = 1" "[because a, b, c "are in H.P." therefore (1)/(a), (1)/(b), (1)/(c) "are in A.P."]`
Thus, the given equation has equal roots each equal to 1.