The differential equation $\mathrm{dy} / \mathrm{dx}=\left[\mathrm{x}\left(1+\mathrm{y}^{2}\right)\right] /\left[\mathrm{y}\left(1+\mathrm{x}^{2}\right)\right]$ represents a family of
1) parabola
2) hyperbola
3) circle
4) ellipse

Hint: Here we are differentiating the given expression and finding the required value. After that, we are checking what kind of equation it is. Parabola, hyperbola.. etc .. each are having their respective general equation.

Complete step by step Solution:
Given,
$\dfrac{d y}{d x}=\dfrac{\left(1+y^{2}\right) x}{y\left(1+x^{2}\right)}$
Taking integral
$\Rightarrow \int \dfrac{2 y}{1+y^{2}} d y=\int \dfrac{2 x}{1+x^{2}} d x$
To determine the functions that will characterize the area, displacement, and volume that result from a combination of small data that cannot be measured separately, integrals are calculated. The concept of limit is employed broadly in calculus whenever algebra and geometry are applied. Limits assist us in analyzing the behavior of points on a graph, such as how they approach one another until their distance approaches zero.
The integration of inverse of a function is called the logarithm of that same function
So the equation becomes
$\Rightarrow \log \left(1+y^{2}\right)=\log \left(1+x^{2}\right)+\log k$
\[\Rightarrow \left( 1+{{y}^{2}} \right)=\left( 1+{{x}^{2}} \right)k\]
This is the equation for hyperbola

Hence, the correct option is 2.

Note: A hyperbola is a collection of points where the separation between each focus is a constant greater than 1. In other words, the locus of a moving point in a plane is such that the ratio of the distance from a fixed point to the distance from a fixed line is always larger than 1.