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-

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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