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Question Number 64112 by mr W last updated on 13/Jul/19

Commented by mr W last updated on 13/Jul/19

$F i n d t h e r a d i u s o f t h e n - t h i n s c r i b e d$ $c i r c l e w i t h i n t h e p a r a b o l a .$
	Answered by ajfour last updated on 13/Jul/19

	$x^{2} + {(y - r_{1})}^{2} = r_{1}^{2}$ $x + \frac{d y}{d x} (y - r_{1}) = 0$ $\frac{d y}{d x} = \frac{x}{r_{1} - y} \Rightarrow \frac{d^{2} y}{{d x}^{2}} = \frac{r_{1} - y + x \frac{d y}{d x}}{{(r_{1} - y)}^{2}}$ $A t o r i g i n w e t h e n h a v e$ $\frac{1}{r_{1}} = 2 \Rightarrow r_{1} = 1 / 2$ $l e t c e n t r e o f n^{t h} c i r c l e b e (0, c_{n})$ $c_{n} = 2 (r_{1} + r_{2} + . . . + r_{n - 1}) + r_{n}$ $l e t t h i s c i r c l e t o u c h p a r a b o l a a t$ $(h, k) .$ $h = r_{n} \sin θ$ $k = c_{n} - r_{n} \cos θ$ $2 r_{n} \sin θ = \tan θ$ $\Rightarrow r_{n} \cos θ = \frac{1}{2}$ $c_{n} - \frac{1}{2} = r_{n}^{2} - \frac{1}{4}$ $\Rightarrow c_{n} = r_{n}^{2} + \frac{1}{4}$ $c_{n} = 2 (r_{1} + r_{2} + . . . + r_{n - 1}) + r_{n}$ $\frac{1}{2} (r_{n}^{2} - r_{n} + \frac{1}{4}) = r_{1} + r_{2} + . . . + r_{n - 1}$ $\frac{1}{2} {(r_{n} - \frac{1}{2})}^{2} = r_{1} + r_{2} + . . . + r_{n - 1}$ $\frac{1}{2} (r_{n + 1} - \frac{1}{2}) = r_{1} + r_{2} + . . + r_{n}$ $\Rightarrow \frac{1}{2} (r_{n + 1} - r_{n}) (r_{n + 1} + r_{n} - 1)$ $= r_{n} - r_{n - 1}$ $\Rightarrow r_{n + 1}^{2} - r_{n + 1} + r_{n} - r_{n}^{2} = 2 r_{n} - 2 r_{n - 1}$ $\Rightarrow r_{n + 1}^{2} - r_{n + 1} = r_{n}^{2} + r_{n} - 2 r_{n - 1}$ $o r {(r_{n + 1} - \frac{1}{2})}^{2} = {(r_{n} + \frac{1}{2})}^{2} - 2 r_{n - 1}$ $w i t h r_{1} = \frac{1}{2} .$
	Commented by mr W last updated on 13/Jul/19

	$t h a n k y o u s i r!$ $a c c . t o d r a w i n g i t s e e m s t h a t$ $r_{n} = n - \frac{1}{2}$ $b u t t h i s {d o e s n}^{'} t m a t c h$ ${(r_{n + 1} - \frac{1}{2})}^{2} = {(r_{n} + \frac{1}{2})}^{2} - 2 r_{n - 1}$ $c a n y o u p l e a s e r e c h e c k ?$
	Answered by mr W last updated on 13/Jul/19

	$f i r s t c i r c l e t o u c h e s t h e p a r a b o l a a t (0, 0)$ $y^{'} = 2 x$ $y^{″} = 2 \Rightarrow R = \frac{1}{2}$ $\Rightarrow r a d i u s o f t h e f i r s t c i r c l e r_{1} = \frac{1}{2}$ $l e t r_{n} = r a d i u s o f n t h c i r c l e$ $d_{n} = d i a m e t e r o f n t h c i r c l e, d_{n} = 2 r_{n}$ $t h e n t h c i r c l e t o u c h e s t h e n - 1 t h c i r c l e$ $a t (0, d_{1} + d_{2} + . . . d_{n - 1})$ $l e t a_{n} = d_{1} + d_{2} + . . . d_{n - 1} = 2 (r_{1} + r_{2} + . . . + r_{n - 1})$ $e q n . o f n t h c i r c l e :$ $x^{2} + {(y - a_{n} - r_{n})}^{2} = r_{n}^{2}$ $i n t e r s e c t i o n w i t h p a r a b o l a y = x^{2} :$ $y + {(y - a_{n} - r_{n})}^{2} = r_{n}^{2}$ $\Rightarrow y^{2} - [2 (a_{n} + r_{n}) - 1] y + a_{n} (a_{n} + 2 r_{n}) = 0$ $d u e t o t a n g e n c y :$ $Δ = {[2 (a_{n} + r_{n}) - 1]}^{2} - 4 a_{n} (a_{n} + 2 r_{n}) = 0$ $4 (a_{n}^{2} + 2 a_{n} r_{n} + r_{n}^{2}) - 4 (a_{n} + r_{n}) + 1 - 4 a_{n}^{2} - 8 a_{n} r_{n} = 0$ $r_{n}^{2} - r_{n} - (a_{n} - \frac{1}{4}) = 0$ $\Rightarrow r_{n} = \frac{1}{2} + \sqrt{a_{n}}$ $\Rightarrow r_{n} = \frac{1}{2} + \sqrt{2 (r_{1} + r_{2} + . . . + r_{n - 1})}$ ${(r_{n} - \frac{1}{2})}^{2} = 2 (r_{1} + r_{2} + . . . + r_{n - 1})$ $\Rightarrow r_{n}^{2} - r_{n} + \frac{1}{4} = 2 (r_{1} + r_{2} + . . . + r_{n - 1}) . . . (i)$ $\Rightarrow r_{n - 1}^{2} - r_{n - 1} + \frac{1}{4} = 2 (r_{1} + r_{2} + . . . + r_{n - 2})$ $\Rightarrow r_{n - 1}^{2} + r_{n - 1} + \frac{1}{4} = 2 (r_{1} + r_{2} + . . . + r_{n - 1}) . . . (i i)$ $(i) - (i i) :$ $(r_{n} + r_{n - 1}) (r_{n} - r_{n - 1} - 1) = 0$ $\Rightarrow r_{n} - r_{n - 1} - 1 = 0$ $\Rightarrow r_{n} - r_{n - 1} = 1$ $\Rightarrow r_{1}, r_{2}, r_{3}, . ., r_{n} a r e A . P . w i t h c o m m o n$ $d i f f e r e n c e 1, s i n c e r_{1} = \frac{1}{2},$ $\Rightarrow r_{n} = \frac{1}{2} + (n - 1) \times 1 = n - \frac{1}{2}$
	Commented by ajfour last updated on 13/Jul/19

	$N i c e t h o u g h t s i r, g e t t i n g r i d o f r_{n - 2}!$ $Y o u h a d s o l v e d i t b e f o r e, t h o u g h$ $n o t t h i s r i g o r o u s w a y .$
	Commented by mr W last updated on 13/Jul/19

	$t o b e h o n e s t i f o u n d a t f i r s t o n l y t h e$ $r e l a t i o n r = \frac{1}{2} + \sqrt{a} a n d d r e w t h e c i r c l e s$ $o n e a f t e r o n e . a f t e r i h a v e p o s t e d t h e q u e s t i o n,$ $i f o u n d t h a t t h e c i r c l e s t o u c h e a c h o t h e r$ $a l w a y s a t p e r f e c t s q u a r e s 1, 4, 9, 16 . . .$ $t h a t m e a n s t h e r a d i i a r e a n A . P .$ $a f t e r i h a v e r e a d y o u r p o s t i t r i e d t o f i n d$ $a w a y t o p r o v e t h i s . i f o u n d a w a y a f t e r$ $s o m e t r i a l s .$
	Commented by Tawa11 last updated on 04/Jan/22

	$I just see this . God bless you sir .$
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