Correct Answer - D
Let there be two stars `1` and `2` as shown below
Let `P` be a point between `C_(1)` and `C_(2)`, where gravitational field strength is zero. Hence `(GM)/(r_(1)^(2))=(G(16M))/(r_(2)^(2)), (r_(2))/(r_(1))=4, r_(1)+r_(2)=10 a`
`:. r_(2)=(4/(4+1))(10a)=8a`
`r_(1)=2a` Now, the body of mass `m` is projected from the surface of large star towards 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`. therefore, the body should be projected to just cross point `P` because beyond that the particle is attracted towards the smaller star itself. from conservation of mechanical energy `1/2 mv^(2)` =potential energy of the body at `P` -potential energy at the surface of larger star.
`:. 1/2mv_(min)^(2)=[(GMm)/(r_(1))-(16GMm)/(r_(2))]-[-(GMm)/(10a-2a)-(16GMm)/(2a)]`
`1/2mv_(min)^(2)=(45/8)(GMm)/a`
`v_(min)=(3sqrt(5))/2(sqrt((GM)/a))`