Correct Answer - A::B::C
Let there are two stars `1` and `2` as shows below.
Let `P` is a point between `C_(1)` and `C_(2)`, where gravitational field strength is zero. Or at `P` field stregth due to star `1` is equal and opposite to the field stength due to star `2`. Hence,
`(GM)/(r_(1)^(2)) = (G(16 M))/(r_(2)^(2))` or `(r_(2))/(r_(1)) = 4`
also `r_(1) + r_(2) = 10a`
`:. r_(2) ((4)/(4 + 1)) (10a) = 8a`
and `r_(1) = 2a`
NOw, the body of mass `m` is projected from the surface of larger star to wards the smaller one. Between `C_(2)` and `P` it is attracted towards `2` and between `C_(1)` and `p` it will be attracted towards `1`. point `P` becouse beyond that the particle is attracted towards the smallar star itself.
From conservation of mechanical energy`(1)/(2) m nu_(min)^(2)`
`=` Potential energy of body at `P`
`- `Potential energy at the surface of the larger star.
`:. (1)/(2) m nu_(min)^(2) = [-(GMm)/(r_(1)) - (16 GMm)/(r_(2))]`
`- [-(GMm)/(10 a - 2a) - (16 GMm)/(2a)]`
`= [-(GMm)/( 2a) - (16 GMm)/(8a)] - [-(GMm)/( 8a) - (16 GMm)/(a)]`
or `(1)/(2) m nu_(min)^(2) = ((84)/(8)) (GMm)/(a)`
`:. nu_(min) = (3sqrt(5))/(2) (sqrt(GM)/(a))`