If the equation x^2 + β^2 = 1 − 2βx and x^2 + α^2 = 1 − 2αx have one and only one root in common, then ∣α − β∣ is equal to


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Question Number 20684 by Tinkutara last updated on 31/Aug/17

$I f t h e e q u a t i o n x^{2} + β^{2} = 1 - 2 β x a n d$ $x^{2} + α^{2} = 1 - 2 α x h a v e o n e a n d o n l y$ $o n e r o o t i n c o m m o n, t h e n ∣ α - β ∣ i s$ $e q u a l t o$
	Answered by $@ty@m last updated on 02/Sep/17

	$L e t t h e c o m m o n r o o t b e p$ $\Rightarrow {(p + α)}^{2} = 1 & {(p + β)}^{2} = 1$ $C a s e - I . p + α = 1, p + β = 1$ $\Rightarrow α = β$ $\Rightarrow b o t h e q u a t i o n s a r e s a m e .$ $C a s e - I I . p + α = 1 & p + β = - 1$ $\Rightarrow α - β = 2$ $C a s e - I I I . p + α = - 1 & p + β = 1$ $\Rightarrow - (α - β) = 2 \Rightarrow (α - β) = - 2$ $C o m b i n i n g t h e r e s u l t s o f C a s e I I$ $a n d C a s e - I I I, w e g e t$ $∣ α - β ∣= 2$
	Commented by Tinkutara last updated on 02/Sep/17

	$Thank you very much Sir!$
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