Let `nge3` and let `C_(1),C_(2),....,C_(n)` be circles witht radii, `r_(1),.r_(2),....,r_(n),` respectively. Assume that `C_(1) and C_(i+1)` touch external for `2leilen-1`. It is also given that the x-axis and the line `y=2sqrt(2)x+10` are tangential to each of the ci rcles. Then `r_(1),r_(2),....,r_(n),` are in-

Options

A. ana arithmetic progeression with common difference `3+sqrt(3)`
B. a geometric progeression with common ratio `3+sqrt(3)`
C. an arithmetic progeression with common difference `2+sqrt(3)`
D. a geometric progeression with common ratio `2+sqrt(3)`

Answer

D

Explanation

`((-5)/(sqrt(2)),0)`
`tantheta=(2-tan theta//2)/(1-tan ^(2)theta//2)`
`2sqrt(2)=(2-tan theta//2)/(1-tan ^(2)theta//2)`
`sqrt(2)tan^(2) theta//2+tan theta-sqrt(2)=0`
`tan theta//2 =(-1+-sqrt(1+8))/(2sqrt(2))`
`=(-1+-3)/(2sqrt(2))=(1)/(sqrt(2))or -sqrt(2)`
`:. tan theta//2=(1)/(sqrt(2))`
`In DeltaOMN sin theta//2=(r_(1))/(ON) sin theta//2 =(1)/(sqrt(2))`
`ON=sqrt(3r_(1))`
In `DeltaOMNsin theta//2=(r_(1))/(ON+r_(1)+r_(2))rArr(1)/(sqrt(3))=(r_(2))/(sqrt(3)r_(1)+r_(1)+r_(2))`
`sqrt(3)r_(1)+r_(1)+r_(2)=sqrt(3)r_(2)`
`r_(1)(sqrt(3)+1)=r_(2)(sqrt(3)-1)`
`(r_(2))/(r_(1))=(sqrt(3)+1)/(sqrt(3)-1)`
`=((sqrt3+1)^(2))/(2)=2+sqrt(3)`