Let two non-collinear unit vectors $\hat { a }$ and $\hat { b }$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow { O P }$ (where $O$ is the origin) is given by $\hat { a } \cos t + \hat { b } \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\overrightarrow { O P }$ and $\hat { u }$ be the unit vector along $\overrightarrow { O P }$. Then, (A) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$ (B) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$ (C) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$ (D) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
Let two non-collinear unit vectors $\hat { a }$ and $\hat { b }$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow { O P }$ (where $O$ is the origin) is given by $\hat { a } \cos t + \hat { b } \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\overrightarrow { O P }$ and $\hat { u }$ be the unit vector along $\overrightarrow { O P }$. Then,\\
(A) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$\\
(B) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$\\
(C) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$\\
(D) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$