The differential equation $\frac{dy}{dx}=\frac{\sqrt{1-y^{2}}}{y}$ determine family of circles with:

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to analyze the given differential equation:

$$
\frac{dy}{dx} = \frac{\sqrt{1-y^2}}{y}
$$

This differential equation suggests a relationship between $ y $ and $ x $ that can be interpreted geometrically. Let's solve this differential equation to understand the family of curves it represents.

### Solving the Differential Equation

1. **Separate Variables:**

We can rewrite the equation as:

$$
   y \, dy = \sqrt{1-y^2} \, dx
   $$

2. **Integrate Both Sides:**

Integrating both sides, we have:

$$
   \int y \, dy = \int \sqrt{1-y^2} \, dx
   $$

The left side integrates to:

$$
   \frac{y^2}{2} + C_1
   $$

For the right side, let $ y = \sin(\theta) $, then $ dy = \cos(\theta) \, d\theta $, and $ \sqrt{1-y^2} = \cos(\theta) $. Thus, the integral becomes:

$$
   \int \cos(\theta) \, d\theta = x + C_2
   $$

Therefore, we have:

$$
   \frac{y^2}{2} = x + C
   $$

where $ C = C_2 - C_1 $.

3. **Rearrange to Find the Family of Curves:**

Rearranging gives:

$$
   y^2 = 2x + 2C
   $$

This equation represents a family of circles. To see this, rewrite it as:

$$
   (x - h)^2 + y^2 = r^2
   $$

Comparing, we see that the equation $ y^2 = 2x + 2C $ can be rewritten as:

$$
   (x - (-C))^2 + y^2 = C^2
   $$

This represents a circle with a fixed radius $ r = 1 $ (since the equation simplifies to $ y^2 = 2x + 2C $ which implies $ r^2 = 1 $) and variable centers along the x-axis, specifically at $ (-C, 0) $.

### Explanation of Options

- **Option 1: Variable radii and fixed centre at (0, 1).**

This is incorrect because the center of the circles is not fixed at (0, 1). The center varies along the x-axis.

- **Option 2: Variable radii and fixed centre at (0, -1).**

This is incorrect because, similar to Option 1, the center is not fixed at (0, -1). The center varies along the x-axis.

- **Option 3: Fixed radius 1 and variables centres along x-axis.**

This is correct because the derived equation $ y^2 = 2x + 2C $ represents circles with a fixed radius of 1 and centers that vary along the x-axis.

- **Option 4: Fixed radius 1 and variables centres along y-axis.**

This is incorrect because the centers of the circles vary along the x-axis, not the y-axis.

Thus, Option 3 is the correct answer, as it accurately describes the family of circles determined by the given differential equation.