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Question Number 52360 by ajfour last updated on 06/Jan/19

Commented by ajfour last updated on 06/Jan/19

$I f θ_{m a x} = 150 ° a n d θ_{m i n} = 75 °, f i n d$ $a a n d b i n t e r m s o f r a d i u s R .$
	Answered by mr W last updated on 06/Jan/19

	Commented by ajfour last updated on 06/Jan/19

	$h o w c o m e t h e p e r p e n d i c u l a r$ $b i s e c t o r S i r ? f o r c e r t a i n ?!$
	Commented by mr W last updated on 06/Jan/19

	$l e t M P = h, A M = M B = d$ $\frac{d}{h} = \tan \frac{θ_{m a x}}{2} = m$ $\Rightarrow d = m h$ $\frac{d}{h + 2 R} = \tan \frac{θ_{m i n}}{2} = n$ $\Rightarrow d = n (h + 2 R)$ $\Rightarrow m h = n (h + 2 R)$ $\Rightarrow h = \frac{2 n R}{m - n}$ $\Rightarrow d = \frac{2 m n R}{m - n}$ $a = \sqrt{d^{2} + {(h + R)}^{2}} = \sqrt{{(\frac{2 m n R}{m - n})}^{2} + {(\frac{2 n R}{m - n} + R)}^{2}}$ $\Rightarrow a = \frac{\sqrt{4 m^{2} n^{2} + {(m + n)}^{2}}}{m - n} \times R$ $\frac{b}{2 d} = \frac{h + R}{a}$ $b = \frac{2 d (h + R)}{a} = \frac{2 \times 2 m n}{\sqrt{4 m^{2} n^{2} + {(m + n)}^{2}}} (\frac{2 n}{m - n} + 1) \times R$ $\Rightarrow b = \frac{4 m n (m + n)}{(m - n) \sqrt{4 m^{2} n^{2} + {(m + n)}^{2}}} \times R$ $w i t h θ_{m a x} = 150 °, θ_{m i n} = 75 °$ $m = \tan \frac{150 °}{2} = \tan 75 ° = \sqrt{3} + 2$ $n = \tan \frac{75 °}{2}$ $m = \frac{2 n}{1 - n^{2}}$ ${m n}^{2} + 2 n - m = 0$ $n = \frac{\sqrt{1 + m^{2}} - 1}{m} = 2 (\sqrt{2 - \sqrt{3}} - 1) + \sqrt{3}$ $m + n = 2 (\sqrt{3} + \sqrt{2 - \sqrt{3}})$ $m - n = 2 (2 - \sqrt{2 - \sqrt{3}})$ $m n = 2 \sqrt{2 + \sqrt{3}} - 1$ $. . .$
	Commented by mr W last updated on 06/Jan/19

	$f o r θ = c o n s t a n t, t h e l o c u s o f p o i n t P$ $i s a c i r c l e t h r o u g h p o i n t s A a n d B .$ $A B i s a c h o r d o f t h i s c i r c l e . s u c h t h a t$ $θ_{m a x} h a p p e n s, t h i s c i r c l e m u s t t a n g e n t$ $t h e c i r c l e w i t h r a d i u s R .$
	Commented by mr W last updated on 06/Jan/19

	Commented by ajfour last updated on 06/Jan/19

	$f i n e S i r, l e t m e c h e c k w i t h b = 0,$ $i n a c o u p l e o f m i n u t e s, p l e a s e . .$
	Commented by mr W last updated on 06/Jan/19

	$i t h i n k i m a d e a m i s t a k e . i t i s c o r r e c t$ $t h a t t h e c i r c l e t h r o u g h A B s h o u l d$ $t a n g e n t t h e c i r c l e w i t h r a d i u s R, b u t$ $A B i s n o t a l w a y s c e n t r i c t o t h e c i r c l e$ $w i t h r a d i u s R . b u t i h a v e a s s u m e d$ $t h e y a r e c e n t r i c t o e a c h o t h e r .$
	Commented by mr W last updated on 06/Jan/19

	$i w a n t t o s a y, m y s o l u t i o n i s c o r r e c t,$ $b u t t h e r e a r e a l s o o t h e r p o s s i b i l i t i e s .$
	Commented by mr W last updated on 06/Jan/19

	$s i n c e b o t h a a n d b a r e t o b e d e t e r m i n e d,$ $i t i s o k t o a s s u m e t h a t A B i s c e n t r i c t o$ $t h e c i r c l e a n d w e g e t t h e s o l u t i o n a s$ $i d i d . b u t i f o n e o f a a n d b i s f i x e d, t h e n i t$ $i s w r o n g t o a s s u m e t h a t A B i s c e n t r i c$ $t o t h e c i r c l e .$
	Commented by ajfour last updated on 06/Jan/19

	$y e s S i r, i t r i e d t o a g r e e b u t m y$ $s k e t c h i n g d i d n o t l e t m e a g r e e t o t a l l y$ $a b o u t t h e p e r p e n d i c u l a r b i s e c t o r$ $l i n e y o u d r e w, n o t g e n e r a l l y .$
	Commented by mr W last updated on 07/Jan/19

	$t h e s i t u a t i o n i s f o l l o w i n g :$ $w h e n a a n d b a r e g i v e n, w e g e t u n i q u e$ $θ_{m a x} a n d θ_{m i n} .$ $b u t w h e n θ_{m a x} a n d θ_{m i n} a r e g i v e n, w e$ $c a n n o t g e t u n i q u e a a n d b . t h e s o l u t i o n$ $i g a v e i s a c o r r e c t s o l u t i o n, b u t n o t t h e$ $o n l y s o l u t i o n .$
	Commented by ajfour last updated on 07/Jan/19

	$o h y e s, S i r . T h a n k y o u s o m u c h!$ $d i n t o c c u r t o m e, w i t h o u t y o u r w o r d s .$
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