The correct option is C Exactly 90∘
Let v1 and v2 be the speeds of the balls after the collision. Then Equation (ii) above gives
v1sinθ1=v2sinθ2 (v)
Also, conservation of kinetic energy gives 12mu2=12mv21+12mv22
or u2=v21+v22 (vi)
Also Equation (i) above becomes
mu=mv1cosθ1+mv2cosθ2
or v2cosθ2=u−v1cosθ1 (vii)
Squaring (v) and (vii) and adding we get v22=u2+v21−2uv1cosθ1 (viii)
Eliminating v2 between (vi) and (vii), we get cosθ1=v2u (ix)
Eliminating v2 between (v) and (vii) we get tanθ2=sinθ1(uv1−cosθ1) (x)
Using (ix) in (x), we get
tanθ2=sinθ1(1cosθ1−cosθ1)=sinθ1cosθ1(1−cos2θ1) =cotθ1=tan(90∘−θ1)
or θ2=90∘−θ1orθ1+θ2=90∘.
Hence the correct choice is (c).