Form the differential equations by eliminating the arbitrary constants from the following equations :
` (1) (x-a)^(2) + y^(2) =1 `

Answer

To form the differential equation by eliminating the arbitrary constant from the given equation \((x-a)^2 + y^2 = 1\), follow these steps:

### Step 1: Differentiate the equation
We start with the equation:
\[ (x - a)^2 + y^2 = 1 \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}((x - a)^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(1) \]
Using the chain rule, we get:
\[ 2(x - a) \cdot \frac{d}{dx}(x - a) + 2y \cdot \frac{dy}{dx} = 0 \]
Since \(\frac{d}{dx}(x - a) = 1\), this simplifies to:
\[ 2(x - a) + 2y \frac{dy}{dx} = 0 \]

### Step 2: Simplify the equation
Dividing the entire equation by 2:
\[ (x - a) + y \frac{dy}{dx} = 0 \]
Rearranging gives:
\[ y \frac{dy}{dx} = - (x - a) \]

### Step 3: Solve for \(a\)
From the original equation, we can express \(a\) in terms of \(x\) and \(y\):
\[ a = x + y \frac{dy}{dx} \]

### Step 4: Substitute \(a\) back into the equation
Now, we substitute this expression for \(a\) back into the equation:
\[ (x - (x + y \frac{dy}{dx}))^2 + y^2 = 1 \]
This simplifies to:
\[ (-y \frac{dy}{dx})^2 + y^2 = 1 \]
Expanding gives:
\[ y^2 \left(\frac{dy}{dx}\right)^2 + y^2 = 1 \]

### Step 5: Factor out \(y^2\)
Factoring out \(y^2\):
\[ y^2 \left(\left(\frac{dy}{dx}\right)^2 + 1\right) = 1 \]

### Step 6: Final form of the differential equation
Thus, the required differential equation is:
\[ y^2 \left(\left(\frac{dy}{dx}\right)^2 + 1\right) = 1 \]