If ` vec b` is a vector whose initial point divides thejoin of `5 hat ia n d5 hat j` in the ratio `k :1` and whose terminal point is the origin and `| vec b|lt=sqrt(37),t h e nk` lies in the interval a. `[-6,-1//6]` b. `(-oo,-6]uu[-1//6,oo)` c. `[0,6]` d. none of these

Options

A. `[-6,-1//6]`
B. `[-oo,-6]cup[-1//6,oo]`
C. `[0,6]`
D. none of these

Answer

B

Explanation

The point the divides `5hati and 5hatj` in th ratio of
`k:1` is `((5hatj)k+(5hati)1)/(k+1)`
`therefore b=(5hati+5khatj)/(k+1)`
also, `|b| le sqrt(37)`

`implies (1)/(k+1)sqrt(25+25k^(2)) le sqrt(37)`
or `5sqrt(1+k^(2)) le sqrt(37)(k+1)`
On squaring both sides, we get
`25(1+k^(2)) lt 37(k^(2)+2k+1)`
or `6k^(2)+37k+6 ge 0` or `(6k+1)(k+6)ge0`
`k in (-oo,-6) cup [-(1)/(6),oo]`.