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

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,
$1 + \left(\text{r}\right)^{2} + 2 \text{r} = 1 \left(\left(\text{r}\right)^{2} + 8\right) + 9$
$\Rightarrow 2\text{r}=16\Rightarrow \text{r}=8$