The centres of two circles and 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 and and be a circle touching circles and externally. If a common tangent to and pas

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 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 tangent 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]

Smart Achievers · Accepted Answer

(8)

$Let r be the radius of required circle.$

$Clearly, in ∆ C_{1} C C_{2}, C_{1} C = C_{2} C = r + 1 and P is mid point of C_{1} C_{2}$

$∴$ $C P ⊥ C_{1} C_{2}, Also P M ⊥ C C_{1}$

$Now ∆ P M C_{1} ~ ∆ C P C_{1} (By AA similarity)$

$∴$ $\frac{M C_{1}}{P C_{1}} = \frac{P C_{1}}{C C_{1}} \Rightarrow \frac{1}{3} = \frac{3}{r + 1}$ $\Rightarrow r + 1 = 9 \Rightarrow r = 8$

Solution

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Solution

Ans. (8) Let r be the radius of required circle. Clearly, in ∆C1CC2, C1C=C2C=r+1 and P is mid point of C1C2 ∴ CP⊥C1C2, Also PM⊥CC1 Now ∆PMC1~∆CPC1 (By AA similarity) ∴ MC1PC1=PC1CC1 ⇒ 13=3r+1⇒r+1=9 ⇒ r=8

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