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 $(0,1)$
• B. variable radii and a fixed centre at $(0,-1)$
• C. fixed radius $1$ and variable centres along the $X$-axis
• D. fixed radius $1$ and variable centres along the $Y$-axis

Answer 1

Answer: (C)

Given, $\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{y}$
$\Rightarrow \int \frac{y}{\sqrt{1-y^2}}dy=\int dx$
$\Rightarrow -\sqrt{1-y^2}=x+c\Rightarrow (x+c{)}^2+y^2=1$
Here, centre $(-c,0)$ and radius $=1$