The figure above shows a unit circle inscribed in the parabloa
. The coordinates of center of the circle inscribed can be written as
, where
are non-negative integers with
coprime.
Find .
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T h e d i a g r a m i s s y m m e t r i c a b o u t y − a x i s ∴ l e t c e n t e r b e ( 0 , a ) ∴ e q . o f c i r c l e ⟶ x 2 + ( y − a ) 2 = 1 s o l v i n g w i t h t h e e q u a t i o n o f t h e p a r a b o l a y = x 2 w e g e t ⟹ x 2 + ( x 2 − a ) 2 = 1 ⟹ x 4 + ( 1 − 2 a ) x 2 + ( a 2 − 1 ) = 0 T h e p a r a b o l a a n d t h e c i r c l e w i l l b e t a n g e n t t o e a c h o t h e r w h e n t h i s e q . i n x 2 h a s r e a l a n d e q u a l r o o t s ⟹ ( 1 − 2 a ) 2 = 4 ( a 2 − 1 ) ⟹ a = 4 5 t h e c e n t e r o f t h e c i r c l e i s ( 0 , 4 5 ) s o a + b + c = 9