Question

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

• A. variable radii and a fixed centre at $(0,0)$
• B. Fixed radius 1 and variable centers along the x-axis
• C. Fixed radius 1 and variable centers along the y-axis
• D. variable radii and a fixed center at $(0,-1)$

Answer 1

The correct option is B Fixed radius 1 and variable centers along the x-axis

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

$\int \frac{y}{\sqrt{1-y^2}}dy=\int dx\Rightarrow -\sqrt{1-y^2}=x+c$

$\Rightarrow (x+c{)}^2+y^2=1$

$\therefore$ centre $=(-c,0);$ radius $=1$