interesting rotational motion problem>!!

sum 1 plz try.......

V_a= -vcosθ i+v sinθ j

V_b= +u cos α i + u sinα j

V_AB= (-vcosθ-u cosα) i + (v sinθ-u sinα) j

ω = (v sinθ-u sinα)/r (since velocity along x-axis gives no angular velocity)

\vec{v}_{a,b}=\vec{v}_{a,g}-\vec{v}_{b,g} = \left(v_a-v_b \right)

So angualr velocity should be given by b).....pls point out my mistake if wrong.

put \theta _{b}=0 , \theta _{a}=\pi

then clearly ang vel = 0

hence options a,b are eliminated , and c,d are still dere

now put \theta _{b}= \theta _{a}=\frac{\pi }{2}

now
\upsilon _{a/b}= v_{a}-v_{b}

since they are in the same direction, eqn is true even for magnitude )

hence ang vel of A relative to B is simply \frac{\upsilon _{a/b}}{r}=\frac{ v_{a}-v_{b}}{r}

but this will be in the clockwise direction , hence in anticlockwise direction it will be \frac{ v_{b}-v_{a}}{r}

and out of c,d c gives correct ans

sorry for my nonconceptual method [4]but i really solved ds in 25 secs [3]