Correct Answer  Option 1 : m
_{1} = m
_{2}
The correct answer is option 1) i.e. m1 = m2
CONCEPT:

The universal law of gravitation: It states that every object in the universe attracts every other with a force varying directly as the product of their masses and inversely as the square of the distance between them.
This is given by:
\(F =\frac{GMm}{R^2}\)
Where F is the gravitational force, M and m are the masses of two objects, R is the distance between the centre of two objects, and G is the universal gravitational constant.
CALCULATION:
Given that:
Mass of one part = m_{1}
Mass of the second part = m_{2} = M  m_{1}
Let us assume the distance between their centres is R.
\(⇒ F =\frac{Gm_1m_2}{R^2} = \frac{Gm_1(Mm_1)}{R^2}\)
For this force to be maximum, \(\frac{dF}{dm_1} = 0\)
\(⇒ \frac{d F}{dm_1} =\frac{d( \frac{Gm_1(Mm_1)}{R^2})}{dm_1}\)
\(⇒ \frac{d F}{dm_1} =\frac{G(M  2m_1)}{R^2}\)
⇒ M  2m_{1} = 0
⇒ m1 = M/2
∴ m_{2 }= M/2
Thus, the gravitational force is maximum when m1 = m2.