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