Lelt two non collinear unit vectors `hata and hatb` form and acute angle. A point P moves so that at any time t the position vector `vec(OP)` (where O is the origin) is given by `hatacost+hatbsint.` When P is farthest from origin O, let M be the length of `vec(OP) and hatu` be the unit vector along `vec(OP)` Then (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)`

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|)`