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?
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
$${what}\:{do}\:{you}\:{mean}\:{with}\:“{the}\:{assumed} \\ $$$${point}''?\:{where}\:{does}\:{this}\:{point}\:{lie}? \\ $$
Commented by mr W last updated on 29/Aug/24
$${if}\:{this}\:{point}\:{lies}\:{in}\:{the}\:{ring}\:{area}, \\ $$$${then}\:{there}\:{are}\:\mathrm{4}\:{circles}\:{which} \\ $$$${tangent}\:{both}\:{circles}\:{and}\:{pass}\:{through} \\ $$$${this}\:{point}. \\ $$
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
$${only}\:{when}\:{the}\:{point}\:{lies}\:{between} \\ $$$${the}\:{two}\:{concentric}\:{circles},\:{as}\:{shown} \\ $$$${in}\:{the}\:{example}\:{above},\:{there}\:{exist} \\ $$$${circles}\:{which}\:{tangent}\:{the}\:{both}\: \\ $$$${circles}. \\ $$
Commented by behi834171 last updated on 29/Aug/24
$${Thank}\:{you}\:{so}\:{much}\:{dear}\:{master}. \\ $$$${if}\:{the}\:{radii}\:{of}\:{cocentric}\:{circles}\:\:{are}\:{given}, \\ $$$${is}\:{it}\:{possible}\:{to}\:{find}\:{the}\:{radii}\:{of}\: \\ $$$${tangent}\:{circles}? \\ $$
Commented by mr W last updated on 29/Aug/24
$${yes}.\:{say}\:{the}\:{radius}\:{of}\:{big}\:{circle}\:{is}\:{R} \\ $$$${and}\:{the}\:{radius}\:{of}\:{small}\:{circle}\:{is}\:{r}, \\ $$$${then}\:{the}\:{radius}\:{of}\:{the}\:{circles}\:{which} \\ $$$${tangent}\:{these}\:{two}\:{circles}\:{is} \\ $$$${r}_{\mathrm{1}} =\frac{{R}−{r}}{\mathrm{2}}\:\left({the}\:{red}\:{ones}\right)\:{or} \\ $$$${r}_{\mathrm{2}} =\frac{{R}+{r}}{\mathrm{2}}\:\left({the}\:{blues}\:{ones}\right) \\ $$