If the roots of the equation a(b - c)x 2+ b(c - a)x + c(a - b) = 0 are equal, then which of the following is true?
2/b = (1/a) + (1/c)
Note that \(a(b-c)+b(c-a)+c(a-b)=0\), so \(x=1\) is a root. For equal roots, both roots are \(1\).
Product of roots: \(\dfrac{c(a-b)}{a(b-c)}=1 \implies c(a-b)=a(b-c)\)
\[ac-bc=ab-ac \implies 2ac=ab+bc\]
Divide by \(abc\): \(\dfrac{2}{b}=\dfrac{1}{a}+\dfrac{1}{c}\). Hence a, b, c are in HP.
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