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Question Number 209712 by mr W last updated on 19/Jul/24

Commented by mr W last updated on 19/Jul/24

$g e n e r a l c a s e$ $O A = a$ $O B = b > a$ $∠ A O B = θ < \frac{π}{2}$ $n = e v e n n u m b e r = 2 k$ $f i n d t h e m i n i m u m o f$ ${A C}_{1} + C_{1} C_{2} + . . . + C_{n - 1} C_{n} + C_{n} B$ $i n t e r m s o f a, b, n, θ .$
Commented by mahdipoor last updated on 19/Jul/24

$i f A = C_{2} = C_{4} = . . . = C_{n} (n = 2 k)$ $a n d C_{1} = C_{3} = . . . = C_{n - 1} = D (O D = O A = a)$ $L = \sum^{} C_{n - 1} C_{n} = 2 . n . a . s i n \frac{θ}{2} + \sqrt{a^{2} + b^{2} - 2 a b . c o s θ}$ $i s m i n i m u m (a < b)$
Commented by mr W last updated on 19/Jul/24

$y o u m e a n$ $L_{m i n} = 2 n a \sin \frac{θ}{2} + \sqrt{a^{2} + b^{2} - 2 a b \cos θ}$ $b u t t h i s i s n o t t r u e . e x a m p l e :$ $a = 2, b = 6, n = 2, θ = 20 °$ $L_{m i n} = 2 \times 2 \times 2 \sin \frac{20 °}{2} + \sqrt{2^{2} + 6^{2} - 2 \times 2 \times 6 \times \cos 20 °}$ $= 8 \sin \frac{20 °}{2} + \sqrt{40 - 24 \cos 20 °} \approx 5.56$ $b u t m i n i m u m = 2 \sqrt{7} \approx 5.29$ $s e e Q 209637$
	Answered by mahdipoor last updated on 20/Jul/24
	${ ((if (n+1)θ≤180 ⇒L_(min) =(√(a^2 +b^2 −2ab.cos((n+1)θ))))),((if (n+1)θ≥180 ⇒L_(min) =a+b)) :} i will try to send photo , that is explain my answer$
	${\begin{cases} i f (n + 1) θ ⩽ 180 \Rightarrow L_{m i n} = \sqrt{a^{2} + b^{2} - 2 a b . c o s ((n + 1) θ)} \\ i f (n + 1) θ ⩾ 180 \Rightarrow L_{m i n} = a + b \end{cases}$ $i will try to send photo, that is explain my answer$
	Answered by mr W last updated on 21/Jul/24

	Commented by mr W last updated on 21/Jul/24

	$t h e s h o r t e s t p a t h f r o m A t o B i s$ $t h e p a t h o f a l i g h t b e t w e e n t w o$ $m i r r o r s .$ $i f t h e l i g h t r a y g o e s d i r e c t l y f r o m$ $A t o B t h e s h o r t e s t p a t h l e n g t h i s$ $L_{m i n} = A B = \sqrt{a^{2} + b^{2} - 2 a b \cos θ}$ $i f t h e l i g h t r a y a r e r e f l e c t e d$ $n = 2 k t i m e s b y t h e m i r r o r s, t h e$ $s h o r t e s t p a t h l e n g t h i s$ $L_{m i n} = {({A C}_{1} + C_{1} C_{2} + . . . + C_{n} B)}_{m i n}$ $= \sqrt{a^{2} + b^{2} - 2 a b \cos (2 k + 1) θ}$ $(s e e a l s o Q 209637)$ $s u c h a l i g h t r a y p a t h i s o n l y p o s s i b l e$ $w h e n (2 k + 1) θ ⩽ 180 °, i . e . k ⩽ \frac{90}{θ} - \frac{1}{2} .$ $t h a t m e a n s,$ $w h e n k ⩽ k_{1} = ⌊ \frac{90}{θ} - \frac{1}{2} ⌋,$ ${({A C}_{1} + C_{1} C_{2} + . . . + C_{n} B)}_{m i n}$ $= \sqrt{a^{2} + b^{2} - 2 a b \cos (2 k + 1) θ}$ $w h e n k > k_{1} = ⌊ \frac{90}{θ} - \frac{1}{2} ⌋,$ ${({A C}_{1} + C_{1} C_{2} + . . . + C_{n} B)}_{m i n}$ $= 2 (k - k_{1}) a \sin θ + \sqrt{a^{2} + b^{2} - 2 a b \cos (2 k_{1} + 1) θ}$ $i n t h i s c a s e w e r e p e a t 2 (k - k_{1}) t i m e s$ $t h e s h o r t e s t d i s t a n c e A D a n d t h e n$ $f o l l o w i n g t h e p a t h o f t h e l i g h t r a y .$
	Commented by mr W last updated on 21/Jul/24

	Commented by mahdipoor last updated on 21/Jul/24

	$i f (2 k + 1) θ ⩾ 180, m y a n s w e r i s :$ $w h e n C_{1} = C_{2} = . . . = C_{n} = O \Rightarrow L = m i n = a + b$ $f o r e x a m p l e : a = 2, b = 6, θ = 45, n = 4 (k = 2)$ $m y a n s : L_{m i n} = 8$ $y o u r a n s : k_{1} = 1 \Rightarrow L_{m i n} = 2 \sqrt{2} + \sqrt{40 + 12 \sqrt{2}} = 10.4$
	Commented by mr W last updated on 21/Jul/24

	$t h e p o i n t s b e t w e e n A a n d B s h o u l d$ $l i e a l t e r n a t e l y o n b o t h l i n e s, t h e r e f o r e$ $p o i n t O s h o u l d b e e x c e p t e d .$
	Commented by mahdipoor last updated on 21/Jul/24

	$i think we must accept point O is on$ $both line, then \forall C_{i} = O alternately on both line$ $if O is excepted; as you sey (C_{i} \neq O)$ $C_{2 i + 1} \in O B, {O C}_{2 i + 1} = v e r y s m a l l = β$ $C_{2 i} \in O A, {O C}_{2 i} = v e r y s m a l l = α$ $L_{m i n} = {[a^{2} + β^{2} - 2 a β . c o s θ]}^{0.5} + {[b^{2} + α^{2} - 2 b α . c o s θ]}^{0.5} +$ $\frac{n}{2} \sqrt{α^{2} + β^{2} - 2 α β . c o s θ}$ $f o r l a s t e x a m p l e (a = 2, b = 6, θ = 45, n = 4)$ $f o r β = α = 1 \Rightarrow L_{m i n} = 8.34 < 10.4$ $f o r β = α = 0.1 \Rightarrow L_{m i n} = 8.01 < 10.4$ $. . .$
	Commented by mr W last updated on 21/Jul/24

	$y o u a r e r i g h t . w e c a n s e l e c t t h e$ $p o i n t s v e r y c l o s e t o p o i n t O .$ $L_{m i n} \approx a - ϵ + \sqrt{ϵ^{2} + b^{2} - 2 ϵ b \cos (2 k_{1} - 1) θ}$ $\to a + b$
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