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  4. If the roots of the equation a(b-c) x2+b(c-a) x+c(a-b)=0 are equal, then a, b, c are in

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Q. If the roots of the equation $a(b-c) x^{2}+b(c-a) x+c(a-b)=0$ are equal, then $a, b, c$ are in

Bihar CECEBihar CECE 2011

Solution:

Since, roots of the equation are equal.
$\therefore $ Discriminant $=0$
i.e., $b^{2}(c-a)^{2}-4 a c(b-c)(a-b)=0$
$\Rightarrow b^{2}(c+a)^{2}-4 a b c(a+c)+4 a^{2} c^{2}=0$
$\Rightarrow (b(a+c)-2 a c)^{2}=0$
$ \Rightarrow b=\frac{2 a c}{a+c}$
$\Rightarrow a, b, c$ are in $HP$.