Question

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the midpoint of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$ , then the radius (in units) of the circle $C$ is

Answer 1

Answer: 8

In $\Delta {\left(\text{CPC}\right)}_2\text{,}{\left(\text{r}+1\right)}^2={\left(\text{α}\right)}^2+3^2$ ...(i)

In $\Delta \text{PTC}$ ,
${\left(\alpha \right)}^2={\left(\text{r}\right)}^2+\left(3^2-1^2\right)$
${\alpha }^2={\text{r}}^2+8$
Putting the value of ${\alpha }^2$ in equation (i), we get,
$1+{\left(\text{r}\right)}^2+2\text{r}=1\left({\left(\text{r}\right)}^2+8\right)+9$
$\Rightarrow 2\text{r}=16\Rightarrow \text{r}=8$