Circle 3 Question 23
23. 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 mid-point 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 tangents to $C _1$ and $C$ passing through $P$ is also a common tangent to $C _2$ and $C$, then the radius of the circle $C$ is …
(2009)
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Solution:
- $(r+1)^{2}=\alpha^{2}+9$
$$ \begin{aligned} & r^{2}+8=\alpha^{2} \\ & \Rightarrow \quad r^{2}+2 r+1=r^{2}+8+9 \\ & \Rightarrow \quad 2 r=16 \Rightarrow r=8 \end{aligned} $$
