What is minimum distance between xy = 4 and x^2 +y^2 = 4 ?


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Question Number 115112 by bemath last updated on 23/Sep/20

$W h a t i s m i n i m u m d i s t a n c e b e t w e e n$ $x y = 4 a n d x^{2} + y^{2} = 4 ?$
	Answered by bobhans last updated on 28/Sep/20

	$y o u w a n t a l i n e {t h a t}^{'} s n o r m a l t o b o t h$ $c u r v e s a n d n a m e l y y = x$ $T h e l i n e i n t e r s e c t t h e h y p e r b o l a a t$ $x = y = \pm 2 a n d t h e c i r c l e a t x = y = \pm \sqrt{2}$ $W e j u s t n e e d t o c o m p u t e t h e d i s t a n c e$ $b e t w e e n (2, 2) a n d (\sqrt{2}, \sqrt{2}) .$ $M i n i m u n d i s t a n c e = \sqrt{{(2 - \sqrt{2})}^{2} + {(2 - \sqrt{2})}^{2}}$ $= \sqrt{2 (6 - 4 \sqrt{2})}$ $= 2 \sqrt{3 - 2 \sqrt{2}} = 2 (\sqrt{2} - 1)$ $= 2 \sqrt{2} - 2$
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