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

(C) fixed radius 1 and variable centres along the $x$-axis

(A) variable radii and a fixed centre at $(0,1)$

(B) variable radii and a fixed centre at $(0,-1)$

(D) fixed radius 1 and variable centres along the $y$-axis

JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Differential Equations

The differential equation

\frac{d y}{d x} = \frac{1 - y ^{2}}{y}

determines a family of circles with

Answer: (C)

Separating variables:

\frac{y}{1 - y ^{2}} d y = d x

Integrating both sides:

\int \frac{y}{1 - y ^{2}} d y = \int d x

Let

u = 1 - y^{2}

, then

d u = - 2 y d y

- \frac{1}{2} \int \frac{d u}{u} = x + c ⟹ - 1 - y^{2} = x + c

Squaring both sides:

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

This represents a family of circles with fixed radius 1 and centres at

(- c, 0)

, which lie along the

x

-axis.

JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Single correct option