A rocket is fired from the earth to moon. The distance between the earth and moon is r and the mass of the earth is 81 times the mass of the moon. The gravitational force on the rocket will be zero when its distance from moon is

Let the rocket be at a distance $x$  from the moon when the gravitational force on it is zero.

Distance of that point from earth $=r-x$

Mass of earth(M_e) = 81 X Mass of the moon (M_m)

Gravitational force on the rocket due to earth $F_e=\frac{Gm{M}_e}{{\left(r-x\right)}^2}$

Gravitational force on rocket due to moon $F_m=\frac{Gm{M}_m}{x^2}$

$F_e=F_m$

$\frac{Gm{M}_e}{{\left(r-x\right)}^2}=\frac{Gm{M}_m}{x^2}$

$\Rightarrow \frac{r-x}{x}=\sqrt{\frac{M_e}{M_m}}=\sqrt{81}=9$

$\Rightarrow \frac{r}{x}-1=9$
$\Rightarrow \frac{r}{x}=10$

$\therefore x=\frac{r}{10}$