A double star is a system of two stars (say masses m₁ and m₂) moving around the centre of inertia of the sustem due to gravitation . Then , ratio of sweeps area of atar of mas m₁ to the star of the mass m₂is .............................

Ans given in book is : m₂ ²/m₁ ²

Let the CM be at a distance r from mass m₁ and distance between the masses be d.
Taking m₁ as the origin
r = (m₁0 + m₂d)/(m₁ + m₂)
=> r = m₂d/(m₁ + m₂)  ----1
Again, d- r = d - m₂d/(m₁ + m₂)
    = m₁d/(m₁ + m₂)  ----2
Centripetal force will be directed towards the centre of mass. From two body problem.
Gm₁m₂/d² = m₁v₁²/r
=>  Gm₂/d² = v₁²/{ m₂d/(m₁ + m₂)}
=>  v₁ = m₂ {G/(m₁ + m₂)d}^1/2  ---3
Similarly orbital velocity of the second planet
 v₂ = m₁ {G/(m₁ + m₂)d}^1/2 ---4.
From 3 and 4
v₁/v₂ = m₂/m₁  ----5.
Area swept by the area vectors are given by
A₁ =  ½r²ω₁
A₂ =  ½(d-r)²ω₂
A₁/A₂ =  r₁²ω₁/ (d-r)²ω₂
  rω₁ = v₁
   and ω₁(d-r) = v₂
=> A₁/A₂ = (rv₁/ (d-r)v₂)
From eqn. 1 and 5.
A₁/A₂ = m₂²/m₁²