{ ((h^2 +y^2 +(k−z)^2 =s^2 )),((a^2 +(b−y)^2 +z^2 =s^2 )),((ah+y(y−b)+z(z−k)=0)),((((h+a)/2)+yz−(b−y)(k−z)=1)),((b+a(k−z)+hz=1)),((k+h(b−y)+ay=1)) :} Find s_(min) or at least express s=f(y) or g(z).


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Question Number 74024 by ajfour last updated on 18/Nov/19
${ ((h^2 +y^2 +(k−z)^2 =s^2 )),((a^2 +(b−y)^2 +z^2 =s^2 )),((ah+y(y−b)+z(z−k)=0)),((((h+a)/2)+yz−(b−y)(k−z)=1)),((b+a(k−z)+hz=1)),((k+h(b−y)+ay=1)) :} Find s_(min) or at least express s=f(y) or g(z).$
${\begin{cases} h^{2} + y^{2} + {(k - z)}^{2} = s^{2} \\ a^{2} + {(b - y)}^{2} + z^{2} = s^{2} \\ a h + y (y - b) + z (z - k) = 0 \\ \frac{h + a}{2} + y z - (b - y) (k - z) = 1 \\ b + a (k - z) + h z = 1 \\ k + h (b - y) + a y = 1 \end{cases}$ $F i n d s_{m i n} o r a t l e a s t e x p r e s s$ $s = f (y) o r g (z) .$
Commented by mind is power last updated on 18/Nov/19

$S_{\min} for the system has solution ?$
	Answered by ajfour last updated on 18/Nov/19

	$h^{2} + k^{2} - 2 k z = s^{2} - (x^{2} + y^{2}) = s^{2} - r^{2}$ $a^{2} + b^{2} - 2 b y = s^{2} - r^{2}$ $a h + r^{2} = b y + k z$ $_________________________$ $\Rightarrow {(h - a)}^{2} + k^{2} + b^{2} = 2 s^{2} . . . (I)$ $\frac{h + a}{2} = 1 + b k - k y - b z . . . (i)$ $a k = 1 - b - (h - a) z . . . (i i)$ $b h = 1 - k + (h - a) y . . . (i i i)$ $\Rightarrow y = \frac{b h + k - 1}{h - a}; z = \frac{1 - b - a k}{h - a}$ $u s i n g (i i)$ $1 + b k - \frac{(h + a)}{2}$ $= \frac{k (b h + k - 1) + b (1 - b - a k)}{h - a} . . (I I)$ $&$ $r^{2} = \frac{{(b h + k - 1)}^{2} + {(1 - b - a k)}^{2}}{{(h - a)}^{2}}$ $a h + r^{2} = \frac{b (b h + k - 1) + k (1 - b - a k)}{h - a}$ $. . . . . . (I I I)$ $a^{2} + b^{2} + r^{2} - s^{2} = \frac{2 b (b h + k - 1)}{h - a}$ $. . . . . . (I V)$ $N o w e q s . (I) \to (I V) c o n t a i n s$ $h, a, k, b, a n d s .$ $s c a n t h e n b e e x p r e s s e d a s a$ $f u n c t i o n o f a n y o n e o f h, a, b, k .$ $s a y s = f (t) .$ $\Rightarrow t h e r e s e e m s t o b e n u m e r o u s$ $w a y s t o m e e t t h e c o n d i t i o n -$ $(s i x o f e i g h t c o r n e r s o f i n n e r c u b e t o u c h$ $o u t e r c u b e f a c e s, o n e e a c h .)$ $F o r s_{m i n} w e i m p o s e \frac{d s}{d t} = 0 .$
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