The centres of two circles `C_(1) and C_(2)` each of unit radius are at a distance of 6 unit 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

Options

A. 4
B. 8
C. 6
D. 3

Answer

B

Explanation

In `deltaA_(1)MP`, we have
`A_(1)P^(2)=A_(1)M^(2)+ PM^(2)rArr PM=sqrt(9-1)=2sqrt(2)`
Clearly, AP is perpendicular to `A_(1)A_(2)`. Let the radius of circle C be r.
Using Pythagoras theorem in triangles `APA_(1)` and `AMP`, we have

`A A_(1)^(2)=AP^(2)+A_(1)P^(2) and AP^(2)=AM^(2)+PM^(2)`
`rArr (r+1)^(2)=AP^(2)=3^(2) and AP^(2)=r^(2)+(2 sqrt(2))^(2)`
`rArr (r+1)^(2)=r^(2)+8+9 rArr r=8`