If the angles of elevation of the top of tower from three collinear points `A`, `B` and `C`, on a line leading to the foot of the tower, are `30^(@)`, `45^(@)` and `60^(@)` respectively, then the ratio , `AB : BC` is

Options

A. `1:sqrt3`
B. `2:3`
C. `sqrt3:1`
D. `sqrt3:sqrt2`

Answer

C

Explanation

Let OP be the tower of height h and let OC =x. In `triangle`'s COP,BOP and AOP, we have

tan `60^@=h/x , tan 45^@ = h/(x+BC)` and `tan 30^@ =h/(x+BC+AB)`
`rArr h=sqrt3x , h=x+BC` and `sqrt3h=x+BC+AB`
`rArr sqrt3x=x +BC` and `sqrt3(x+BC)=x=BC+AB`
`rArr (sqrt3-1)x = BC` and `(sqrt3-1)x =BC(1-sqrt3)+AB`
`rArr BC=BC (1-sqrt3)+AB`
`rArr sqrt3BC =AB`
`rArr AB:BC=sqrt3:1`