Two circles are assumed that have the same center. How many circles are there that are tangent to both circles and pass through the assumed point?


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Question Number 211127 by behi834171 last updated on 28/Aug/24

Two circles are assumed that have the same center. How many circles are there that are tangent to both circles and pass through the assumed point?
Commented by mr W last updated on 29/Aug/24

$w h a t d o y o u m e a n w i t h ‘ ‘ t h e a s s u m e d$ ${p o i n t}^{″} ? w h e r e d o e s t h i s p o i n t l i e ?$
Commented by mr W last updated on 29/Aug/24

$i f t h i s p o i n t l i e s i n t h e r i n g a r e a,$ $t h e n t h e r e a r e 4 c i r c l e s w h i c h$ $t a n g e n t b o t h c i r c l e s a n d p a s s t h r o u g h$ $t h i s p o i n t .$
Commented by mr W last updated on 29/Aug/24

Commented by behi834171 last updated on 29/Aug/24

Thank you for your attention,sir mr.W. The point can be anywhere: inside the any one of two concentric circles,or,on the side of the outer circle,or outside both circles.
Commented by mr W last updated on 29/Aug/24

$o n l y w h e n t h e p o i n t l i e s b e t w e e n$ $t h e t w o c o n c e n t r i c c i r c l e s, a s s h o w n$ $i n t h e e x a m p l e a b o v e, t h e r e e x i s t$ $c i r c l e s w h i c h t a n g e n t t h e b o t h$ $c i r c l e s .$
Commented by behi834171 last updated on 29/Aug/24

$T h a n k y o u s o m u c h d e a r m a s t e r .$ $i f t h e r a d i i o f c o c e n t r i c c i r c l e s a r e g i v e n,$ $i s i t p o s s i b l e t o f i n d t h e r a d i i o f$ $t a n g e n t c i r c l e s ?$
Commented by mr W last updated on 29/Aug/24

$y e s . s a y t h e r a d i u s o f b i g c i r c l e i s R$ $a n d t h e r a d i u s o f s m a l l c i r c l e i s r,$ $t h e n t h e r a d i u s o f t h e c i r c l e s w h i c h$ $t a n g e n t t h e s e t w o c i r c l e s i s$ $r_{1} = \frac{R - r}{2} (t h e r e d o n e s) o r$ $r_{2} = \frac{R + r}{2} (t h e b l u e s o n e s)$
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