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question mark

if (1/x) + (1/x-a) + (1/x-b)=0.

prove that one root lies between a/3 and 2a/3 , and other lies between b/3 and 2b/3

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A B , 13 Years ago

Grade 9

1 Answers

Jitender Singh

Ans:
Hello Student,
Please find answer to your question below

$\frac{1}{x} + \frac{1}{x-a} + \frac{1}{x-b} = 0$
$\frac{1}{x-a} + \frac{1}{x-b} = \frac{-1}{x}$
$x(2x-a-b) = -(x^{2}-(a+b)x+ab)$
$3x^2 - 2(a+b)x+ab = 0$
$x = \frac{2(a+b)\pm \sqrt{4(a+b)^2-12ab}}{6}$
$x = \frac{(a+b)\pm \sqrt{(a+b)^2-3ab}}{3}$
From here it is clear that one root lie b/w a/3 & 2a/3 and other b/w b/3 & 2b/3.

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