The degree of the differential equation of the curve `(x-a)^(2) + y^(2) =16` will be

Options

A. 0
B. 2
C. 3
D. 1

Answer

To find the degree of the differential equation of the curve \((x-a)^{2} + y^{2} = 16\), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ (x - a)^{2} + y^{2} = 16 \]
We differentiate both sides with respect to \(x\).

### Step 2: Apply the differentiation
Using the chain rule, we differentiate:
\[ \frac{d}{dx}[(x - a)^{2}] + \frac{d}{dx}[y^{2}] = \frac{d}{dx}[16] \]
This gives us:
\[ 2(x - a) \cdot \frac{d}{dx}(x - a) + 2y \cdot \frac{dy}{dx} = 0 \]
Since \(a\) is a constant, \(\frac{d}{dx}(x - a) = 1\). Thus, we have:
\[ 2(x - a) + 2y \frac{dy}{dx} = 0 \]

### Step 3: Simplify the equation
Now, we can simplify this equation by dividing everything by 2:
\[ (x - a) + y \frac{dy}{dx} = 0 \]
Rearranging gives us:
\[ y \frac{dy}{dx} = - (x - a) \]

### Step 4: Express \(a\) in terms of \(x\) and \(\frac{dy}{dx}\)
From the previous equation, we can express \(a\):
\[ a = x + y \frac{dy}{dx} \]

### Step 5: Substitute \(a\) back into the original equation
Now we substitute \(a\) back into the original equation:
\[ (x - (x + y \frac{dy}{dx}))^{2} + y^{2} = 16 \]
This simplifies to:
\[ (-y \frac{dy}{dx})^{2} + y^{2} = 16 \]

### Step 6: Expand and simplify
Expanding gives:
\[ y^{2} \left(\frac{dy}{dx}\right)^{2} + y^{2} = 16 \]
Factoring out \(y^{2}\):
\[ y^{2} \left(\left(\frac{dy}{dx}\right)^{2} + 1\right) = 16 \]

### Step 7: Identify the degree of the differential equation
The degree of a differential equation is defined as the highest exponent of the highest order derivative present in the equation. Here, the highest order derivative is \(\frac{dy}{dx}\) and its highest exponent is 2.

Thus, the degree of the differential equation is:
\[ \text{Degree} = 2 \]

### Final Answer
The degree of the differential equation of the curve \((x-a)^{2} + y^{2} = 16\) is **2**.