Which one of the following is the differential equation of family to circles having centre at the origin ?

Options

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

Answer

To find the differential equation of the family of circles with their center at the origin, we can follow these steps:

### Step 1: Write the equation of the circle
The general equation of a circle with its center at the origin is given by:
\[ x^2 + y^2 = r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Differentiate the equation with respect to \( x \)
We will differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(r^2) \]
Since \( r \) is a constant (the radius), the derivative of \( r^2 \) is 0. Using the chain rule on the left side, we get:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 3: Rearranging the equation
We can rearrange the equation to isolate the terms involving \( dy/dx \):
\[ 2y \frac{dy}{dx} = -2x \]
Dividing through by 2 gives:
\[ y \frac{dy}{dx} = -x \]

### Step 4: Multiply by \( dx \)
To express this in a more useful form, we can multiply both sides by \( dx \):
\[ y \, dy = -x \, dx \]

### Step 5: Rearranging to standard form
Now we can rearrange this equation:
\[ x \, dx + y \, dy = 0 \]

### Conclusion
Thus, the differential equation representing the family of circles centered at the origin is:
\[ x \, dx + y \, dy = 0 \]