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 given by \((x-a)^{2} + y^{2} = 16\), we will follow these steps:

### Step 1: Differentiate the equation with respect to \(x\)

Given the equation:
\[ (x-a)^{2} + y^{2} = 16 \]

We will differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}((x-a)^{2}) + \frac{d}{dx}(y^{2}) = \frac{d}{dx}(16) \]

### Step 2: Apply the differentiation

Using the chain rule, we differentiate:
\[ 2(x-a) \cdot \frac{d}{dx}(x-a) + 2y \cdot \frac{dy}{dx} = 0 \]
Since \(\frac{d}{dx}(x-a) = 1\), we simplify:
\[ 2(x-a) + 2y \frac{dy}{dx} = 0 \]

### Step 3: Rearranging the equation

We can simplify this equation by dividing through by 2:
\[ (x-a) + y \frac{dy}{dx} = 0 \]

### Step 4: Isolate \(\frac{dy}{dx}\)

Rearranging gives us:
\[ y \frac{dy}{dx} = -(x-a) \]
Now, we can express \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{x-a}{y} \]

### Step 5: Identify the highest order derivative and its power

In our expression for \(\frac{dy}{dx}\), we see that the highest order derivative is \(\frac{dy}{dx}\), and it appears to the power of 1. However, we need to consider the original equation's structure.

### Step 6: Formulate the differential equation

From the rearranged equation, we can square both sides to eliminate the fraction:
\[ \left(\frac{dy}{dx}\right)^2 = \frac{(x-a)^2}{y^2} \]

This gives us a differential equation in the form:
\[ \left(\frac{dy}{dx}\right)^2 + \frac{y^2}{16} = 1 \]

### Step 7: Determine the degree

The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made polynomial in derivatives. In this case, the highest order derivative is \(\frac{dy}{dx}\), and it is squared in the equation.

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**.

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