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Question Number 211058 by mr W last updated on 27/Aug/24

Commented by mr W last updated on 27/Aug/24

$f i n d t h e r a d i u s R o f t h e l a r g e s t s p h e r e$ $w h i c h c a n b e p l a c e d b e t w e e n t h e r e d$ $a n d b l u e p l a n e s i n t h e f i r s t o c t a n t .$ $a s s u m e 0 < p ⩽ a, 0 < q ⩽ b, 0 < r ⩽ c .$
	Answered by mr W last updated on 29/Aug/24

	Commented by mr W last updated on 28/Aug/24

	$t h e l a r g e s t s p h e r e a b o v e a n d t o u c h i n g$ $t h e l o w e r p l a n e \frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1 :$ $s a y i t s r a d i u s i s R_{o}, t h e n i t s c e n t e r$ $i s a t (R_{o}, R_{o}, R_{o}) .$ $w e h a v e$ $\frac{\frac{R_{o}}{p} + \frac{R_{o}}{q} + \frac{R_{o}}{r} - 1}{\sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}} = R_{o}$ $\Rightarrow R_{o} = \frac{1}{\frac{1}{p} + \frac{1}{q} + \frac{1}{r} - \sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}}$
	Commented by mr W last updated on 29/Aug/24

	Commented by mr W last updated on 28/Aug/24

	$t h e l a r g e s t s p h e r e u n d e r t h e$ $u p p e r p l a n e \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 :$ $s a y i t s r a d i u s i s R_{i}, t h e n i t s c e n t e r$ $i s a t (R_{i}, R_{i}, R_{i}) .$ $w e h a v e$ $\frac{\frac{R_{i}}{a} + \frac{R_{i}}{b} + \frac{R_{i}}{c} - 1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}} = - R_{i}$ $\Rightarrow R_{i} = \frac{1}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}}$
	Commented by mr W last updated on 28/Aug/24

	Commented by mr W last updated on 29/Aug/24

	${l e t}^{'} s i g n o r e t h e l o w e r p l a n e a t f i r s t .$ $a s s h o w n p r e v i o u s l y, t h e l a r g e s t$ $s p h e r e u n d e r t h e u p p e r p l a n e h a s$ $t h e r a d i u s R_{i} a n d a n d i t s c e n t e r i s$ $a t (R_{i}, R_{i}, R_{i}) .$ $R_{i} = \frac{1}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}}$ $i f t h e d i s t a n c e f r o m (R_{i}, R_{i}, R_{i}) t o$ $t h e l o w e r p l a n e i s e q u a l t o o r l a r g e r$ $t h a n R_{i}, i . e . i f$ $\frac{\frac{R_{i}}{p} + \frac{R_{i}}{q} + \frac{R_{i}}{r} - 1}{\sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}} ⩾ R_{i}, i . e . i f$ $\frac{1}{\frac{1}{p} + \frac{1}{q} + \frac{1}{r} - \sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}} ⩽ R_{i}$ $i . e . i f R_{o} ⩽ R_{i},$ $t h a t m e a n s i f$ $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} - \sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}$ $⩾ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}$ $t h e n t h e l o w e r p l a n e i s b e n e a t h$ $t h e s p h e r e w i t h r a d i u s R_{i} a n d t h i s$ $s p h e r e i s a l s o t h e l a r g e s t s p h e r e$ $b e t w e e n b o t h p l a n e s . t h a t m e a n s$ $i n t h i s c a s e$ $R = R_{i} = \frac{1}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}}$ $i f R_{o} > R_{i}, i . e . i f$ $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} - \sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}$ $< \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}$ $t h e n t h e l a r g e s t s p h e r e b e t w e e n$ $b o t h p l a n e s i s t h e m a x i m u m s p h e r e$ $f r o m f o l l o w i n g t h r e e c a s e s :$ $c a s e 1 :$ $c e n t e r a t (u, R_{1}, R_{1}), r a d i u s R_{1}$ $\frac{\frac{u}{a} + \frac{R_{1}}{b} + \frac{R_{1}}{c} - 1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}} = - R_{1}$ $\Rightarrow u = a - {a R}_{1} (\frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}})$ $\Rightarrow u = a - {a R}_{1} (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}) + R_{1}$ $\Rightarrow u = a - \frac{{a R}_{1}}{R_{i}} + R_{1}$ $\frac{\frac{u}{p} + \frac{R_{1}}{q} + \frac{R_{1}}{r} - 1}{\sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}} = R_{1}$ $\Rightarrow u = p - {p R}_{1} (\frac{1}{q} + \frac{1}{r} - \sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}})$ $\Rightarrow u = p - {p R}_{1} (\frac{1}{p} + \frac{1}{q} + \frac{1}{r} - \sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}) + R_{1}$ $\Rightarrow u = p - \frac{{p R}_{1}}{R_{o}} + R_{1}$ $p - \frac{{p R}_{1}}{R_{o}} + R_{1} = a - \frac{{a R}_{1}}{R_{i}} + R_{1}$ $\frac{R_{1}}{R_{i}} (1 - \frac{p}{a} \times \frac{R_{i}}{R_{o}}) = 1 - \frac{p}{a}$ $\Rightarrow R_{1} = \frac{1 - \frac{p}{a}}{1 - \frac{p}{a} \times \frac{R_{i}}{R_{o}}} \times R_{i}$ $u = a (1 - \frac{R_{1}}{R_{i}}) + R_{1}$ $\Rightarrow u = a (1 - \frac{1 - \frac{p}{a}}{1 - \frac{p}{a} \times \frac{R_{i}}{R_{o}}}) + R_{1} > R_{1}$ $s i m i l a r l y$ $c a s e 2 :$ $c e n t e r a t (R_{2}, v, R_{2}), r a d i u s R_{2}$ $\Rightarrow R_{2} = \frac{1 - \frac{q}{b}}{1 - \frac{q}{b} \times \frac{R_{i}}{R_{o}}} \times R_{i}$ $\Rightarrow v = b (1 - \frac{1 - \frac{q}{b}}{1 - \frac{q}{b} \times \frac{R_{i}}{R_{o}}}) + R_{2} > R_{2}$ $c a s e 3 :$ $c e n t e r a t (R_{3}, R_{3}, w), r a d i u s R_{3}$ $\Rightarrow R_{3} = \frac{1 - \frac{r}{c}}{1 - \frac{r}{c} \times \frac{R_{i}}{R_{o}}} \times R_{i}$ $\Rightarrow w = c (1 - \frac{1 - \frac{r}{c}}{1 - \frac{r}{c} \times \frac{R_{i}}{R_{o}}}) + R_{3} > R_{3}$ $s i n c e \frac{R_{i}}{R_{o}} < 1 a n d 0 < \frac{p}{a}, \frac{q}{b}, \frac{r}{c} ⩽ 1$ $a n d f (x) = \frac{1 - x}{1 - k x} i s a s t r c t l y$ $d e c r e a s i n g f u n c t i o n f o r 0 < k < 1 a n d$ $0 ⩽ x ⩽ 1, w e g e t$ $R = m a x (R_{1}, R_{2}, R_{3})$ $= \frac{1 - m i n (\frac{p}{a}, \frac{q}{b}, \frac{r}{c})}{1 - m i n (\frac{p}{a}, \frac{q}{b}, \frac{r}{c}) \times \frac{R_{i}}{R_{o}}} \times R_{i}$
	Commented by mr W last updated on 27/Aug/24

	$s u m m a r y :$ $w i t h$ $R_{i} = \frac{1}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}}$ $R_{o} = \frac{1}{\frac{1}{p} + \frac{1}{q} + \frac{1}{r} - \sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}}}}$ $i f R_{o} ⩽ R_{i}, t h e n$ $R = R_{i}$ $i f R_{o} > R_{i}, t h e n$ $R = \frac{1 - m i n (\frac{p}{a}, \frac{q}{b}, \frac{r}{c})}{1 - m i n (\frac{p}{a}, \frac{q}{b}, \frac{r}{c}) \cdot \frac{R_{i}}{R_{o}}} \cdot R_{i}$
	Commented by mr W last updated on 27/Aug/24

	$f o r m u l a u s e d :$ $d i s t a n c e f r o m p o i n t t o p l a n e i n s p a c e$
	Commented by mr W last updated on 27/Aug/24

	Commented by ajfour last updated on 27/Aug/24

	$i m m e n s e! i w i l l l o o k c a r e f u l l y . g e t$ $t h e i d e a v e r y w e l l . g r e a t t h i n g f o r$ $q u e s t i o n t r e a s u r e .$
	Commented by mr W last updated on 27/Aug/24

	$t h a n k s f o r c h e c k i n g s i r!$
	Commented by mr W last updated on 27/Aug/24

	$e x a m p l e 1 :$ $u p p e r p l a n e \frac{x}{6} + \frac{y}{9} + \frac{z}{10} = 1$ $l o w e r p l a n e \frac{x}{5} + \frac{y}{3} + \frac{z}{4} = 1$ $a = 6, b = 9, c = 10$ $p = 5, q = 3, r = 4$ $\frac{p}{a} = \frac{5}{6}, \frac{q}{b} = \frac{3}{9} = \frac{1}{3}, \frac{r}{c} = \frac{4}{10} = \frac{2}{5}$ $\Rightarrow m i n = \frac{q}{b} = \frac{1}{3}$ $R_{i} = \frac{1}{\frac{1}{6} + \frac{1}{9} + \frac{1}{10} + \sqrt{\frac{1}{6^{2}} + \frac{1}{9^{2}} + \frac{1}{10^{2}}}}$ $= \frac{102 - 3 \sqrt{406}}{25} \approx 1.662$ $R_{o} = \frac{1}{\frac{1}{5} + \frac{1}{3} + \frac{1}{4} - \sqrt{\frac{1}{5^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}}}}$ $= \frac{47 + \sqrt{769}}{24} \approx 3.114$ $R_{o} > R_{i}$ $R = \frac{1 - \frac{1}{3}}{1 - \frac{1}{3} \times \frac{24 \times (102 - 3 \sqrt{406})}{25 \times (47 + \sqrt{769})}} \times \frac{102 - 3 \sqrt{406}}{25}$ $\approx 1.348$
	Commented by behi834171 last updated on 27/Aug/24

	$m a t h m a t i c i s v e r y b e a u t i f u l .$ $y o u r Q^{'} s & A^{'} s, a r e m o r e t h a n$ $b e a u t i f u l!$ $T h e y a r e a m a z i n g s i r m r W .$ $T h a n k s g o d f o r b e i n g o f y o u i n t h i s f o r u m .$
	Commented by mr W last updated on 27/Aug/24

	$t h a n k s b a c k t o y o u, f a t h e r f r o m B e h i,$ $f o r y o u r a p p r e c i a t i o n a n d s u p p o r t!$
	Commented by necx122 last updated on 27/Aug/24

	$M r . B e h i i t s b e e n a r e a l l y l o n g t i m e . F e e l s$ $g o o d t o s e e y o u r p o s t h e r e a g a i n .$
	Commented by ajfour last updated on 27/Aug/24

	$y e s s o g l a d f o r y o u r r e m a r k b s c k o n t h i s f o r u m .$
	Commented by behi834171 last updated on 28/Aug/24

	$R e d h e a r t t o y o u d e a r m r . W$
	Commented by behi834171 last updated on 28/Aug/24

	$T h a n k y o u s o m u c h s i r w i t h r e d h e r a t!$
	Commented by behi834171 last updated on 28/Aug/24

	$s o h a p p y f o r s e e y o u a g a i n s i r a j f o u r$ $a n d r e d h e a r t t o y o u!$
	Commented by BHOOPENDRA last updated on 01/Sep/24

	$g r e a t M r . W$
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