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The centers of two circles C1 and C2 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 centers of C1 and C2 and C be a circle touching circles C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then radius of the circle C is ? a)4 b)6 c) 8 d)3 Please reply fast.

The centers of two circles C1 and C2 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 centers of C1 and C2 and C be a circle touching circles C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then radius of the circle C is ? a)4 b)6 c) 8 d)3 Please reply fast.

Grade:12

1 Answers

Shaswata Biswas
132 Points
4 years ago
Draw the line joining the centres if C1 and C2. Now put P in the middle such that PC1=PC2= 3 unit. And R1=R2= 1 unit. Since P lies on the tangent if circle C, it must lie on the circle. And the required circle also touches the others externally. The circle C is either above the line joining C1C2 ir below it. Now two acoording to the given conditions, the two tangents pass through point P. Let the centre of the circle be at C with radius R. Now join CC1, CC2 and CP. Here, CC1=(1+R), PC1= 3 unit, CP=R, and angle CPC1=90. Then using Pythgorus Theorem the value of R can be found.The correct answer is 4 unit.

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