Question

The differential equation dy/dx=√1-y2/y determines a family of circles with

• A. (A) Variable radius and fixed centre at (0,1)
• B. (B) Variable radius and fixed centere at (0,-1)
• C. (C) Fixed radius of 1 Unit and variable centre along the X-axis
• D. (D) Fixed radius of 1 Unit and variable centre along the X- axis

Answer 1

Answer: (D)

(D) Fixed radius of 1 Unit and variable centre along the X- axis

Answer 2

The given differential equation is:

dy/dx = √(1-y^2)/y

We can write this equation in the form: dy/√(1-y^2) = dx/y

Integrating both sides: arcsin(y) = ln|x| + C Where C is the constant of integration. Solving for y: y = sin(ln|x| + C) This is the general solution of the differential equation.

We can observe that this solution represents a family of curves which are circles centered on the x-axis.

To see this, we can rewrite the solution as: y = sin(ln|x| + C) = (e^(ln|x|+C) - e^-(ln|x|+C))/2 Simplifying: y = (x - 1/x)/2

This is the equation of a circle centered at (0,0) with radius 1/2.

Therefore, the correct option is (D) fixed radius of 1 unit and variable center along the x-axis.