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Question Number 46611 by ajfour last updated on 29/Oct/18

Commented by ajfour last updated on 29/Oct/18

$A r i n g o f r a d i u s R s t a n d s u p r i g h t$ $a t G (a, b, 0) . A l o n g w h i c h l i n e, i n$ $x y - p l a n e a n d p a s s i n g t h r o u g h G,$ $a s a x i s s h o u l d t h e r i n g b e t i l t e d$ $a n d b y h o w m u c h (θ) s u c h t h a t$ $i t c o m e s i n c o n t a c t w i t h x z, a n d$ $y z p l a n e s i m u l t a n e o u s l y ?$
	Answered by MrW3 last updated on 29/Oct/18

	Commented by MrW3 last updated on 30/Oct/18

	$Δ A B C r e p r e s e n t s t h e p l a n e w h i c h t h e$ $c i r c l e l i e s i n . T h e c i r c l e i s t h e i n s c r i b e d$ $c i r c l e o f t h e t r i a n g l e Δ A B C .$ $A (a, 0, 0)$ $B (0, b, 0)$ $C (0, 0, c)$ $w e n e e d t o d e t e r m i n e a, b a n d c .$ $G (h, k, 0)$ $l e t ∠ O A B = α$ $\Rightarrow A G = \frac{k}{\sin α}, G B = \frac{h}{\cos α}$ $A B = A G + G B$ $\Rightarrow A B = \frac{k}{\sin α} + \frac{h}{\cos α}$ $\tan \frac{A}{2} = \frac{R}{A G} = \frac{R \sin α}{k}$ $\tan \frac{B}{2} = \frac{R}{G B} = \frac{R \cos α}{h}$ $A C = A P + P C = A G + \frac{R}{\tan \frac{C}{2}} = \frac{k}{\sin α} + \frac{R}{\tan \frac{π - A - B}{2}}$ $A C = \frac{k}{\sin α} + R \tan (\frac{A}{2} + \frac{B}{2}) = \frac{k}{\sin α} + R \frac{\frac{R \sin α}{k} + \frac{R \cos α}{h}}{1 - \frac{R^{2} \sin α \cos α}{h k}}$ $\Rightarrow A C = \frac{k}{\sin α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}$ $B C = B Q + Q C = G B + P C$ $\Rightarrow B C = \frac{h}{\cos α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}$ ${A B}^{2} = a^{2} + b^{2}$ ${A C}^{2} = a^{2} + c^{2}$ ${B C}^{2} = b^{2} + c^{2}$ $\Rightarrow a^{2} + b^{2} + c^{2} = \frac{{A B}^{2} + {A C}^{2} + {B C}^{2}}{2}$ $\Rightarrow a^{2} = \frac{{A B}^{2} + {A C}^{2} - {B C}^{2}}{2} = \frac{{[\frac{k}{\sin α} + \frac{h}{\cos α}]}^{2} + {[\frac{k}{\sin α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2} - {[\frac{h}{\cos α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2}}{2} . . . (i)$ $\Rightarrow b^{2} = \frac{{A B}^{2} - {A C}^{2} + {B C}^{2}}{2} = \frac{{[\frac{k}{\sin α} + \frac{h}{\cos α}]}^{2} - {[\frac{k}{\sin α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2} + {[\frac{h}{\cos α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2}}{2} . . . (i i)$ $\Rightarrow c^{2} = \frac{- {A B}^{2} + {A C}^{2} + {B C}^{2}}{2} = \frac{- {[\frac{k}{\sin α} + \frac{h}{\cos α}]}^{2} + {[\frac{k}{\sin α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2} + {[\frac{h}{\cos α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2}}{2} . . . (i i i)$ $o n t h e o t h e r s i d e w e h a v e$ ${A C}^{2} - {(A B \cos α)}^{2} = {B C}^{2} - {(A B \sin α)}^{2}$ ${A C}^{2} - {B C}^{2} = {A B}^{2} (\cos^{2} α - \sin^{2} α) = \cos 2 α {A B}^{2}$ $\Rightarrow {[\frac{k}{\sin α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2} - {[\frac{h}{\cos α} + \frac{R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}]}^{2} = \cos 2 α {[\frac{k}{\sin α} + \frac{h}{\cos α}]}^{2}$ $\Rightarrow [\frac{k}{\sin α} + \frac{h}{\cos α} + \frac{2 R^{2} (h \sin α + k \cos α)}{h k - R^{2} \sin α \cos α}] [\frac{k}{\sin α} - \frac{h}{\cos α}] = \cos 2 α {[\frac{k}{\sin α} + \frac{h}{\cos α}]}^{2}$ $\Rightarrow (\frac{2 h k}{R^{2}} + \sin 2 α) (\frac{k}{h} - \tan α) = (\frac{2 h k}{R^{2}} - \sin 2 α) (\frac{k}{h} + \tan α) \cos 2 α . . . (i v)$ $l e t t = \tan α, μ = \frac{k}{h}, λ = \frac{R^{2}}{h k}$ $\Rightarrow (μ - λ) t^{3} - t^{2} + μ t + (μ λ - 1) = 0 . . . (v)$ $f r o m (v) w e g e t α, w i t h t h i s α w e c a n$ $g e t a, b a n d c f r o m (i) t o (i i i) .$ $l e t d = a l t i t u d e f r o m O t o A B$ $\frac{1}{2} d \times A B = \frac{1}{2} a b \Rightarrow d = \frac{a b}{A B} = \frac{a b}{\frac{k}{\sin α} + \frac{h}{\cos α}} = \frac{a b \sin α \cos α}{k \cos α + h \sin α}$ $\tan θ = \frac{d}{c}$ $\Rightarrow θ = \tan^{- 1} \frac{a b \sin α \cos α}{c (k \cos α + h \sin α)}$
	Commented by MrW3 last updated on 30/Oct/18

	Commented by ajfour last updated on 30/Oct/18

	$GREAT s o l u t i o n S i r,$ $T h a n k s!$
	Commented by MrW3 last updated on 30/Oct/18

	Commented by MrW3 last updated on 30/Oct/18

	$a j f o u r s i r : c a n y o u c o n f i r m t h e r e s u l t$ $v i a o t h e r m e t h o d ? I h a v e a q u e s t i o n$ $f o r e x a m p l e : w i t h a g i v e n p o i n t G (h, k)$ $w h a t i s t h e r a n g e f o r R s u c h t h a t t h e$ $a c t i o n i s p o s s i b l e ?$
	Commented by MrW3 last updated on 30/Oct/18

	Commented by MrW3 last updated on 30/Oct/18

	$R_{0} = \sqrt{h^{2} + k^{2}}$ $i f R = R_{0} t h e r i n g c a n o n l y s t a n d u p r i g h t$ $a n d c a n n o t b e t i l t e d t o w a r d s t h e w a l l s .$ ${(h - R_{m i n})}^{2} + {(k - R_{m i n})}^{2} = R_{m i n}^{2}$ $R_{m i n}^{2} - 2 (h + k) R_{m i n} + (h^{2} + k^{2}) = 0$ $\Rightarrow R_{m i n} = h + k \pm \sqrt{2 h k}$ $\Rightarrow R_{m i n} = h + k - \sqrt{2 h k}$ $\Rightarrow R_{m a x} = h + k + \sqrt{2 h k}$ $c a s e 1 : R_{m i n} ⩽ R ⩽ R_{0}$ $t h e r i n g c a n b e t i l t e d t o w a r d s$ $t h e w a l l s a n d t o u c h b o t h .$ $c a s e 2 : R_{0} < R ⩽ R_{m a x}$ $t h e r i n g c a n t o u c h t h e w a l l s o n l y f r o m$ $o u t s i d e .$
	Commented by ajfour last updated on 31/Oct/18

	$I {c o u l d n}^{'} t f i n d a n y g o o d w a y, S i r$ $a n d w a s b u s y u n e n t a n g l i n g m y$ $h a n d s!$
	Commented by MrW3 last updated on 31/Oct/18

	$t h a n k y o u f o r r e s p o n d i n g s i r! t h i s i s$ $a v e r y n i c e a n d c h a l l e n g i n g q u e s t i o n .$
	Commented by ajfour last updated on 31/Oct/18

	$i t h i n k t h i s i s s a m e a s p r e v i o u s$ $o n e, S i r . B y t h e w a y y o u w e r e$ $t o o p r a g m a t i c i n s u g g e s t i n g t o$ $o b t a i n \tan θ = \frac{d}{c}, S i r .$
	Answered by MrW3 last updated on 01/Nov/18

	Commented by MrW3 last updated on 01/Nov/18

	$a n o t h e r (b e t t e r) w a y t o s o l v e :$ $t h e p r o j e c t i o n o f t h e c i r c l e i n t h e$ $x y - p l a n e i s a n e l l i p s e w h i c h h a s t h e$ $s e m i - m a j o r a x i s R a n d t a n g e n t s$ $x - a x i s a n d y - a x i s s i m u l t a n e o u s l y .$ $a s s u m e t h e s e m i - m i n o r a x i s o f e l l i p s e$ $i s d .$ $i n c o o r d i n a t e s y s t e m x^{'} y^{'} t h e e q n .$ $o f e l l i p s e i s$ $\frac{x^{2}}{R^{2}} + \frac{y^{2}}{d^{2}} = 1$ $G A = \frac{k}{\sin α}, G B = \frac{h}{\cos α}$ $e q n . o f l i n e O A i n x^{'} y^{'} s y s t e m i s$ $\frac{y - d}{x - \frac{k}{\sin α}} = \tan α$ $\Rightarrow \sin α x - \cos α y + d \cos α - k = 0$ $s i n c e {i t}^{'} s a t a n g e n t l i n e o f e l l i p s e,$ $R^{2} \sin^{2} α + d^{2} \cos^{2} α - {(d \cos α - k)}^{2} = 0$ $\Rightarrow R^{2} \sin^{2} α - k^{2} + 2 d k \cos α = 0 . . . (i)$ $e q n . o f l i n e O B i n x^{'} y^{'} s y s t e m i s$ $\frac{y - d}{x + \frac{h}{\cos α}} = - \frac{1}{\tan α}$ $\Rightarrow \cos α x + \sin α y + h - d \sin α = 0$ $s i n c e {i t}^{'} s a t a n g e n t l i n e o f e l l i p s e,$ $R^{2} \cos^{2} α + q^{2} \sin^{2} α - {(h - d \sin α)}^{2} = 0$ $\Rightarrow R^{2} \cos^{2} α - h^{2} + 2 d h \sin α = 0 . . . (i i)$ $(i) + (i i) :$ $R^{2} - k^{2} - h^{2} + 2 d (k \cos α + h \sin α) = 0$ $\Rightarrow d = \frac{k^{2} + h^{2} - R^{2}}{2 (k \cos α + h \sin α)} . . . (i i i)$ $p u t t h i s i n t o (i i) :$ $R^{2} \cos^{2} α + \frac{h (k^{2} + h^{2} - R^{2}) \sin α}{k \cos α + h \sin α} = h^{2}$ $\Rightarrow {(\frac{R}{h})}^{2} \frac{1}{1 + \tan^{2} α} + \frac{1 + {(\frac{k}{h})}^{2} - {(\frac{R}{h})}^{2}}{1 + (\frac{k}{h}) \frac{1}{\tan α}} = 1 . . . (i v)$ $o r w i t h t = \tan α, λ = \frac{R}{h}, μ = \frac{k}{h}$ $\Rightarrow (μ^{2} - λ^{2}) t^{3} - μ t^{2} + μ^{2} t + μ (λ^{2} - 1) = 0 . . . (i v)$ $f r o m (i v) w e c a n g e t α a t f i r s t .$ $t h e n w e g e t d f r o m (i i i) .$ $\sin θ = \frac{d}{R} = \frac{k^{2} + h^{2} - R^{2}}{2 R (k \cos α + h \sin α)}$ $\Rightarrow θ = \sin^{- 1} \frac{1 + {(\frac{k}{h})}^{2} - {(\frac{R}{h})}^{2}}{2 (\frac{R}{h}) [\sin α + (\frac{k}{h}) \cos α]}$ $e x a m p l e s :$ $h = 4, k = 3, R = 4 :$ $\Rightarrow α = 0 °, 29.4346 °$ $\Rightarrow θ = 22.024 °, 14.225 °$ $h = 3, k = 3, R = 2.5 :$ $\Rightarrow α = 30.8212 °, 45 °, 59.1788 °$ $\Rightarrow θ = 34.842 °, 33.635 °, 34.842 °$
	Commented by ajfour last updated on 01/Nov/18

	$Y o u d i d i t S i r . T h i s i s q u i t e$ $b e t t e r . THANKS a g a i n .$
	Commented by ajfour last updated on 02/Nov/18

	$I h a v e p o s t e d a n a l t e r n a t e w a y, S i r .$
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