What-is-minimum-distance-between-xy-4-and-x-2-y-2-4-

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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