We may infer from the question that the mercury column's air expands in volume under isothermal conditions. This implies that for every operation in the system, the product of applied pressure and volume remains constant. With the ultimate height of the mercury column equalised the length by which the mercury column drops may be calculated by equating the beginning and final pressure and volume conditions.

Formula Used:
Barometric formula: $P=\rho gh$
Boyle's Law: $P_2{V}_2=P_1{V}_1$

We are aware that a barometer is a tool used to gauge the local atmospheric pressure. An inverted glass tube is placed in a mercury bath, and when air pressure presses down on the surface of the mercury, part of the mercury rises up the tube. This is how it works. Greater pressures cause the mercury to climb higher.
The barometer is calibrated in accordance with the following formula, which connects the height of mercury in the tube to the pressure applied, for a height h of mercury in the tube:
P = gh, where is the barometer fluid's density (in this case, mercury), g is gravity's acceleration, and h is the height of the mercury column within the glass tube.

Now, when air is added to the area above the mercury level in the upside-down glass tube, the pressure pulls the mercury downward and causes it to sink.
Consequently, we have originally

$P_1=\rho g\left(75\right)$ and $V_1=50-2=48cc$
And finally,
$P_2=\rho g\left(h\right)$and $V_2=50cc$
The barometer registers no change in the ambient temperature throughout this isothermal process, which results in a reduction (increase) in the volume of mercury (air) in the tube with an increase in pressure.

This indicates that a continuous relationship between pressure and volume is found. It is provided by Boyle's law: $PV=$ constant
$\Rightarrow P_1{V}_1=P_2{V}_2\Rightarrow P_2=\frac{P_1{V}_1}{V_2}$
$\Rightarrow \rho gh=\frac{\rho g\left(75\right)\times 48}{50}\Rightarrow h=\frac{75\times 48}{50}=72 \mathrm{cm}$
As a result, the mercury column's length shortens by $75-72=3 \mathrm{cm}$

Hence, the correct answer is option A.