Let two non-collinear unit vector ` hat aa n d hat b` form an acute angle. A point `P` moves so that at any tgiem `t ,` the position vector `O P(w h e r eO` is the origin`)` is given by ` hat acott+ hat bsintdotW h e nP` is farthest from origin `O ,` let `M` be the length of `O Pa n d hat u` be the unit vector along `O Pdot` Then ` hat u=( hat a+ hat b)/(| hat a+ hat b|)a n dM=(1+ hat adot hat b)^(1//2)` ` hat u=( hat a- hat b)/(| hat a- hat b|)a n dM=(1+ hat adot hat b)^(1//2)` ` hat u=( hat a+ hat b)/(| hat a+ hat b|)a n dM=(1+2 hat adot hat b)^(1//2)` ` hat u=( hat a- hat b)/(| hat a- hat b|)a n dM=(1+2 hat adot hat b)^(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|)`