The differential equation $\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{y}$ determine a family of circles with

fixed radius 1 and variable centre along the x-axis

The correct answer is option  (3) : fixed radius 1 and variable centre along the x-axis

The given differential equation is

$\frac{dy}{dx}= \frac{\sqrt{1-y^2}}{y}$

$⇒\frac{-y}{\sqrt{1-y^2}}dy =-dx$

$⇒\sqrt{1-y^2}=-x+C$

$⇒1-y^2 = (x-C)^2 ⇒(x-C)^2+y^2=1$

Clearly, it represents a family of circles of fixed radius 1 and variable centre (C, 0) along the x-axis.