Explanation

(b)
Given, $\frac{dy}{dx} = \frac{\left( 1 + y^{2} \right)x}{y(1 + x^{2})}$
$\Rightarrow$ $\int_{}^{}\frac{2y}{1 + y^{2}}dy = \int_{}^{}\frac{2x}{1 + x^{2}}dx$
$\Rightarrow$ $\log{(1 + y^{2})} = \log{(1 + x^{2})} + \log k$
$\Rightarrow$ $\left( 1 + y^{2} \right) = \left( 1 + x^{2} \right)k$
This equation represents a family of hyperbola.