The differential equation of the family of curves represented by the equation `x^2+y^2=a^2`, is

Options

A. `x+y (dy)/(dx)=0`
B. `y (dy)/(dx)=x`
C. `y(d^2y)/(dx^2) + ((dy)/(dx))^2=0`
D. `x+y("dy"/"dx")^2 =0`

Answer

To find the differential equation of the family of curves represented by the equation \( x^2 + y^2 = a^2 \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the curve:
\[ x^2 + y^2 = a^2 \]
We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2) \]
Since \( a \) is a constant, the derivative of \( a^2 \) is 0. Therefore, we have:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ 2y \frac{dy}{dx} = -2x \]
Dividing both sides by 2:
\[ y \frac{dy}{dx} = -x \]

### Step 3: Express the differential equation
Now, we can express the differential equation:
\[ y \frac{dy}{dx} + x = 0 \]
This is the required differential equation of the family of curves represented by \( x^2 + y^2 = a^2 \).

### Final Answer
The differential equation of the family of curves represented by the equation \( x^2 + y^2 = a^2 \) is:
\[ y \frac{dy}{dx} + x = 0 \]