Let two non-collinear unit vectors `veca and vecb` form an acute angle. A point P moves so that at any time t, time position vector, `vec(OP)` ( where O is the origin) is given by `hata cot t + hatb sin t`. When p is farthest fro origing o, let M be the length of `vec(OP) and hatu` be the unit vector along `vec(OP)` .then

Options

A. `,hatu = (hata+hatb)/(|hata + hatb|) and M = (1 + hata.hatb)^(1//2)`
B. `,hatu = (hata-hatb)/(|hata - hatb|) and M = (1 + hata.hatb)^(1//2)`
C. `hatu = (hata+hatb)/(|hata + hatb|) and M = (1 + 2hata.hatb)^(1//2)`
D. `,hatu = (hata-hatb)/(|hata - hatb|) and M = (1 + 2hata.hatb)^(1//2)`

Answer

a

Explanation

`|vec(OP)|= |hata cos t +hatb sin t |`
`= (cos^(2)t + sin^(2) t + 2cos t tin t hata.hatb) ^(1//2)`
` (1 + 2 cos t tin t hata.hatb)^(1//2)`
` (1+sin 2t hata.hatb)^(1//2)`
`|vec(OP)|_(max)= (1+hata.hatb)^(1//2)"when " t=pi//4`
`hatu = (hata+hatb)/(sqrt2(|hata+hatb|)/sqrt2)= (hata+hatb)/(|hata+hatb|)`