Let non-collinear unit vectors â 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 a $\operatorname{cost}+\hat{b} \sin t$. When $P$ is farthest from origin O , let M be the length of $\overrightarrow{\mathrm{OP}}$ and u be the unit vector along $\overrightarrow{\mathrm{OP}}$. Then,
Select the correct option:
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 $M=(1+2 \hat{a} \cdot \hat{b})^{\frac{1}{2}}$
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