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//16]`
B. `(-oo, -6]uu[-1//6, oo)`
C. `[0, 6]`
D. none of these

Answer

B

Explanation

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

`rArr (1)/(k+1) sqrt(25 + 25k^(2)) le sqrt(37)`
or `5sqrt(1+k^(2)) le sqrt( 37) (k+1)`
Squaring both sides, we get
`" "25( 1+k^(2) ) le 37(k^(2) + 2k+1)`
or ` 6k^(2) + 37k + 6 ge 0 or ( 6k+1) ( k+6) ge 0`
`k in ( -oo, -6] uu [ - (1)/(6) oo)`