Differential Equations 1 Question 7
7. The differential equation $\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}$ determines a family of circles with
(2007, 3M)
(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
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Answer:
Correct Answer: 7. (c)
Solution:
- Given, $\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}$
$$ \begin{array}{ll} \Rightarrow & \int \frac{y}{\sqrt{1-y^{2}}} d y=\int d x \\ \Rightarrow & -\sqrt{1-y^{2}}=x+c \quad \Rightarrow \quad(x+c)^{2}+y^{2}=1 \end{array} $$
Here, centre $(-c, 0)$ and radius $=1$
