Two bodies of masses $${M_1}$$ and $${M_2}$$ are placed at a distance $$d$$ apart. What is the potential at the position where the gravitational field due to them is zero?

Solution :
Let the gravitational field be zero at a point distant $$x$$ from $${M_1}.$$
$$\eqalign{ & \frac{{G{M_1}}}{{{x^2}}} = \frac{{G{M_2}}}{{{{\left( {d - x} \right)}^2}}};\frac{x}{{d - x}} = \sqrt {\frac{{{M_1}}}{{{M_2}}}} \cr & x\sqrt {{M_2}} = \sqrt {{M_1}} d - x\sqrt {{M_1}} \cr & x\left[ {\sqrt {{M_1}} + \sqrt {{M_2}} } \right] = \sqrt {{M_1}} d \cr & x = \frac{{d\sqrt {{M_1}} }}{{\sqrt {{M_1}} + \sqrt {{M_2}} }},d - x = \frac{{d\sqrt {{M_2}} }}{{\sqrt {{M_1}} + \sqrt {{M_2}} }} \cr} $$
Potential at this point due to both the masses will be
$$\eqalign{ & - \frac{{G{M_1}}}{x} - \frac{{G{M_2}}}{{\left( {d - x} \right)}} \cr & = - G\left[ {\frac{{{M_1}\left( {\sqrt {{M_1}} + \sqrt {{M_2}} } \right)}}{{d\sqrt {{M_1}} }} + \frac{{{M_2}\left( {\sqrt {{M_1}} + \sqrt {{M_2}} } \right)}}{{d\sqrt {{M_2}} }}} \right] \cr & = - \frac{G}{d}{\left( {\sqrt {{M_1}} + \sqrt {{M_2}} } \right)^2} \cr & = - \frac{G}{d}\left( {{M_1} + {M_2} + 2\sqrt {{M_1}} \sqrt {{M_2}} } \right) \cr} $$