Show that if the earth attracts two bodies $A$ and $B$ placed at the same distance from the centre of the earth with the same force,then their masses are equal.

(N/A) Let $m_{1}$ and $m_{2}$ be the masses of the two bodies and $M$ be the mass of the earth.
Let the distance of the two bodies from the centre of the earth be $r$.
According to Newton's law of universal gravitation,the force $F$ exerted by the earth on a body of mass $m$ at a distance $r$ is given by $F = \frac{GMm}{r^{2}}$,where $G$ is the universal gravitational constant.
For body $A$,the force is $F_{1} = \frac{GMm_{1}}{r^{2}}$.
For body $B$,the force is $F_{2} = \frac{GMm_{2}}{r^{2}}$.
Given that the forces are equal,$F_{1} = F_{2}$.
Therefore,$\frac{GMm_{1}}{r^{2}} = \frac{GMm_{2}}{r^{2}}$.
By cancelling the common terms $\frac{GM}{r^{2}}$ from both sides,we get $m_{1} = m_{2}$.
Thus,the masses of the two bodies are equal.