First slide

Theory of equations

Question

If the quadratic equation $a (b - c) x^{2} +$ $b (c - a) x + c (a - b) = 0,$ where $a, b, c$ are distinct real numbers and $a b c \neq 0$ . has equal roots, then a, b, c are in

Moderate

A

A.P.
B

G.P.
C

H.P.
D

A.G.P.

Solution

As 1 is a root of the equation and equation has equal roots, the other root must be 1. Thus we have
$\begin{aligned} 1 = product of roots = \frac{c (a - b)}{a (b - c)} \\ \Rightarrow a (b - c) = c (a - b) \\ \Rightarrow b = \frac{2 a c}{a + c} \end{aligned}$
$\Rightarrow$ $a, b, c$ are in H.P.

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