First slide

Theory of equations

Question

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

Moderate

A

real and distinct roots
B

has equal roots
C

no real root
D

none

Solution

We know that $a (b - c) + b (c - a) + c (a - b) = 0$
So $x = 1$ is a root of the given equation
Let $α$ be the other root.
Then $α \times 1 = \frac{c (a - b)}{a (b - c)} \Rightarrow α = \frac{c (a - b)}{a (b - c)} = \frac{\frac{1}{b} - \frac{1}{a}}{\frac{1}{c} - \frac{1}{b}}$
$\Rightarrow α = 1$ $[∵ a, b, c$ are in H.P $∴ \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P]
Thus, the given equation has equal roots.

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