Three circles `C_(1),C_(2),C_(3)` with radii `r_(1),r_(2),r_(3)(r_(1)ltr_(2)ltr_(3))` respectively are given as `r_(1)=2`, and `r_(3)=8` they are placed such that `C_(2)` lines to the right of `C_(1)` and touches it externally `C_(3)` lies ot the right of `C_(2)` and touches it externally. There exist two stratight lines each of whic is a direct common tangent simultaneously to all the three circles then `r_(2)` is equal to

Options

A. `r_(2)=4`
B. `r_(2)=5`
C. `r_(2)=10`
D. `r_(2)=16`

Answer

A

Explanation

`DE=AK=sqrt((r_(2)=r_(1))^(2)-(r_(2)-r_(1))^(2))=2sqrt(r_(1)r_(2))`
Similarly `EF=2sqrt(r_(2)r_(3))`
`/_APD=theta` and `PD=x` then
`tantheta=(AD)/(PD)=(BE)/(PE)=(CF)/(PF)=(r_(1))/x`
`(r_(3))/(x+2sqrt(r_(1)r_(2)))=(r_(3))/(x+2sqrt(2r_(1)r_(2))+sqrt(r_(2)r_(3)))`
Hence `(r_(2)-r_(1))/(2sqrt(r_(1)r_(2)))=(r_(3)-r_(2))/(2sqrt(r_(2)r_(3)))`
`implies r_(2)=sqrt(r_(1)r_(3)impliesr_(2)sqrt(2xx8)=4`